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Mathematical modelling of the unbending of continuously cast steel slabs Uehara, Masatsugu 1983

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C  MATHEMATICAL MODELLING OF THE UNBENDING OF CONTINUOUSLY CAST STEEL SLABS by MASATSUGU UEHARA M.S.University  Of  Tokyo,1976  A THESIS SUBMITTED IN P A R T I A L FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES Metallurgical  We a c c e p t  this  Engineering  t h e s i s as c o n f o r m i n g  to the r e q u i r e d  standard  THE UNIVERSITY OF B R I T I S H COLUMBIA A u g u s t 1983  ©  M a s a t s u g u U e h a r a , 1983  I  In p r e s e n t i n g  this thesis i n partial  f u l f i l m e n t o f the  r e q u i r e m e n t s f o r an advanced degree a t the of  B r i t i s h Columbia, I agree that  it  freely available  University  the L i b r a r y  f o r reference and study.  s h a l l make I further  agree that permission f o r e x t e n s i v e copying o f t h i s  thesis  f o r s c h o l a r l y p u r p o s e s may be g r a n t e d b y t h e h e a d o f my department o r by h i s o r her r e p r e s e n t a t i v e s . u n d e r s t o o d .that c o p y i n g o r p u b l i c a t i o n o f t h i s  It is thesis  f o r f i n a n c i a l g a i n s h a l l n o t b e a l l o w e d w i t h o u t my w r i t t e n permission.  Department  of  Meta./lu,r^ic6,/ Etplneer'irt^  The U n i v e r s i t y o f B r i t i s h 1956 Main M a l l V a n c o u v e r , Canada V6T 1Y3 Date  Ouj. /P**  ,  Columbia  /pr3  ABSTRACT  A two-dimensional,elasto-plastic,finite-element has  been  developed  deformation  of  slab during  to  calculate  a partially  the  solidified  elastic  two-dimensional  model  calculations.  plane-stress The  effects  in part  of  by s h i f t i n g  internal  cracks  calculations radius)  the  a  that  plant  one-point  internal  i n the upper higher  cracks for  of  sufficient motion  points  data.  A  that a  for  the  have been  i n two s t e p s .  comparing  bending  shell  casting  cracks  predictions  From  the  bow-type  results  of of  caster(l0.5m  appear  at the s o l i d i f i c a t i o n  due t o s t r a i g h t e n i n g o f t h e s t r a n d  speeds.  The c r i t i c a l  was t a k e n t o be 0.25-0.3% a t a s t r a i n  low-carbon  deform  separately  shifted  fronts strain,£  by  around t h e i r  to within the  ,in  strain  at  for internal  r a t e o f 1x10""  s"  1  steels.  I t has been found t h a t  are  is  steel  f o r c a s t i n g s p e e d s o f 1.0,1.2 a n d 1.6m/min, i t h a s been  verified front  with  cast  analysis revealed  the r o l l by  bulging  c a s t i n g machine.  solid-shell  The m o d e l was c h e c k e d  and  continuously  s t r a i g h t e n i n g on a c u r v e d - m o u l d  preliminary,three-dimensional  considered  bending  model  15mm  of  roll-friction  the  the  upper  individual the  lower  which  solidification  Therefore  region  shells  n e u t r a l axes,  respective  force.  low-ductility  and  close  the to  bending the  solidification than as  the a  value  result  resultant internal front is  front  p r e d i c t e d by of  the  as  e  affects  having  low  neutral-axis  T  the  The  £  + e  strain  i t i s important  surface  close  by  theory.  enough t o the  However,  to  among t h e s e  significantly; suppress small  variables  the and  the  roll  cause  solidification  Thus,  t e m p e r a t u r e s and  0.3%  E  u  to  about  bulging s t r a i n , B , t h e  large  correlation  = (2 - 5) e total  s m a l l , lower  w i t h the  streaks)  upper s h e l l .  cracks  one  e  follows;  internal  very  s t r a i n , x ,becomes  total  the  be  interaction  cracks(radial  of  strain  can  to  bulging prevent  bulging  pitches  by  during  straightening. By one-point the  comparing machine r a d i i  bending  bow-type  s m a l l machine r a d i u s of  normal front  c a s t i n g speeds the exceeds the  critical  of  8m,10.5m a n d  c a s t e r , i t has 8.0m  is  tensile value  for crack  at  for  been v e r i f i e d  unfavorable  strain  13m  the  because  a  that at  solidification  formation.  iv  TABLE OF CONTENTS Page Abstract Table of Contents L i s t of Tables L i s t of F i g u r e s L i s t of Symbols Acknowledgement  i i iv vi v i i x i i xiv  Chapter 1  INTRODUCTION  1  2  PREVIOUS WORK AND OBJECTIVES OF PRESENT WORK ..  4  2.1 I n t e r n a l cracks  4  2.2 P r e v i o u s and  cast slabs  .  work on s t r e s s a n a l y s i s of bending  bulging  2.3 O b j e c t i v e s 3  in continuously  6 of p r e s e n t  work  9  BENDING/UNBENDING STRESS ANALYSIS OF CONTINUOUSLY CAST SLABS 3.1 I n t r o d u c t i o n  11 11  3.2 M e c h a n i c a l p r o p e r t i e s of low-carbon s t e e l s a t e l e v a t e d temperature  13  3.2.1 Types of s t r e s s - s t r a i n curves  14  3.2.2 M e c h a n i c a l p r o p e r t y  16  data  3.3 Model development  22  3.3.1 Comparison of the t h r e e - d i m e n s i o n a l and two-dimensional models  23  3.3.2 E f f e c t s of c r e e p i n c a l c u l a t i o n s of b u l g i n g  29  3.3.3 Two-dimensional e l a s t o - p l a s t i c ELEMENT  34  Finite  3.3.4 Boundary c o n d i t i o n s 3.3.4.1 R o l l 3.3.4.2 S h i f t  friction  36 force"  of the boundary c o n d i t i o n  38 .  40  V  3.3.5  C a l c u l a t i o n flow  45  4  CALCULATION CONDITIONS  49  5  MODEL PREDICTIONS AND D I S C U S S I O N  63  5.1 R e s u l t s  63  5.1.1  of c a l c u l a t i o n s  Comparison of model p r e d i c t i o n w i t h plant data  5.1.2 B u l g i n g 5.1.3  71  strain  73  Bending/Unbending  5.2 C o r n e r  strain  and c r a c k  strain formation  5.3 C r e e p e f f e c t s on t h e c r i t i c a l 6  73 88  strain  94  CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK ...  98  6. 1 C o n c l u s i o n s  98  6.2 S u g g e s t i o n s  f o r f u t u r e work  102  REFERENCES  103  A ppendix I Mechanical p r o p e r t i e s adopted in.the bulging c a l c u l a t i o n f o r comparison with the experimental r e s u l t s of. M o r i t a  108  II" III IV V VI  D e r i v a t i o n of t h e F i n i t e - E l e m e n t e l a s t o - p l a s t i c problems M a t e r i a l m a t r i x [D] (plane the F i n i t e Element  equations  109  s t r e s s ) used i n 111  T h i c k - w a l l e d c y l i n d e r under pressure (plane s t r a i n )  internal  E s t i m a t i o n of r o l l (upper s h e l l )  force  friction  Results of c a l c u l a t i o n ( C a s e 2 t o Case 10)  f o r the  113 i n Case 1 117  o f bending and b u l g i n g 118  vi  LIST OF TABLES Table I Studies II Measured  Page of c r i t i c a l data  strain  of b u l g i n g  for i n t e r n a l cracks  by M o r i t a  ...  5 31  2 6  III C a l c u l a t i o n c o n d i t i o n s f o r unbending of continuously cast slabs  53  IV S t r a i n s at s o l i d i f i c a t i o n f r o n t on the c e n t e r plane normal to the wide face  70  V Maximum b u l g i n g d e f l e c t i o n between No.1 and No.2 r o l l s VI Bending and b u l g i n g  s t r a i n at s o l i d i f i c a t i o n  85 front  95  vii  LIST OF FIGURES Figure 1  Page Schematic drawing of d i f f e r e n t types of c a s t i n g machines"  2  0  2 3.1  Elongation bending  at the s o l i d i f i c a t i o n  front  during 8  S t r e s s - s t r a i n curves f o r a u s t e n i t i c i r o n at e l e v a t e d temperatures and low s t r a i n r a t e s "  15  3.2  Assumed elevated  19  3.3  Strain-hardening exponent Zener-Hollomon parameter  3.4  I n f l u e n c e of s t r a i n - h a r d e n i n g exponent s t r e s s - s t r a i n curve '  3.5  Schematic diagram of the t h r e e - d i m e n s i o n a l f i n i t e - e l e m e n t mesh f o r the bending a n a l y s i s  25  3.6  Predicted d i s t o r t i o n s of the s l a b three-dimensional finite-element analysis  26  1  3.7  mechanical p r o p e r t i e s temperature as  of  slab  at  a f u n c t i o n of on the  by the bending  20 21  Predicted x x - s t r a i n d i s t r i b u t i o n i n the cross section of the s l a b by the t h r e e - d i m e n s i o n a l f i n i t e - e l e m e n t bending a n a l y s i s  27  Comparison of bending s t r a i n s between (a)Three-dimensional and (b)Two-dimensional model  28  Comparison of maximum b u l g i n g p r e d i c t e d by the c r e e p model and e l a s t o - p l a s t i c m o d e l .  30  3.10  Schematic diagram c f the two-dimensional f i n i t e - e l e m e n t mesh f o r the b u l g i n g a n a l y s i s  31  3.11  Comparison of b u l g i n g s t r a i n s p r e d i c t e d by the plane s t r e s s and plane strain finiteelement a n a l y s i s  33  Influence of the mesh s i z e on b u l g i n g s t r a i n in the e l a s t o - p l a s t i c f i n i t e - e l e m e n t a n a l y s i s  33  3.8  3.9  2 2  3.12  viii  Figure 3.13  3.14 3.15  Page Schematic diagram of the boundary conditions adopted i n the two-dimensional f i n i t e - e l e m e n t bending a n a l y s i s  37  I n f l u e n c e of c o e f f i c i e n t of r o l l the r e s u l t a n t bending s t r a i n  41  f r i c t i o n on  C o e f f i c i e n t of r o l l f r i c t i o n of hot r o l l i n g as a f u n c t i o n of temperature 3 2  3.16  P r e d i c t e d bending bending model  3.17  P r e d i c t e d bending and b u l g i n g one-step bending model  3.18  Flow c h a r t f o r the c a l c u l a t i o n and b u l g i n g s t r a i n  3.19 4.1  s t r a i n with the one-step  43  s t r a i n with the  temperature  and  44  of the bending 47  Flow c h a r t f o r the c a l c u l a t i o n of the bending and b u l g i n g s t r a i n ( " e l a s t o - p l a s t i c r o u t i n e " ) Surface  42  48  s h e l l thickness in  the continuous c a s t i n g of s l a b 4.2  R o l l p r o f i l e of the 10.5m  4.3  R o l l p r o f i l e of the 8.0m  4.4  R o l l p r o f i l e of the 13.0m  4.5  Assumed s t r e s s - s t r a i n curves f o r the s l a b i n Case 1 , Assumed s t r e s s - s t r a i n curves f o r the s l a b i n Case 2,3,6,7,8,9 and 10  59  Assumed s t r e s s - s t r a i n curves f o r the s l a b Case 4  in ..  60  curves f o r the s l a b i n ..  61  4.6 4.7 4.8 4.9  Assumed Case 5  stress-strain  radius caster  51 ....  radius caster radius caster  Schematic diagram of the two-dimensional finite-element mesh f o r the bending bulging analysis  52 54  ....  and  55 58  62  ix  Figure 5.1 5.2 5.3  Page Predicted distortion b u l g i n g i n Case 1  due t o b e n d i n g  and 64  Predicted x x - s t r a i n c o n t o u r s due t o b e n d i n g and b u l g i n g i n Case 1  65  P r e d i c t e d x y - s t r a i n c o n t o u r s due and b u l g i n g i n Case 1  66  to  bending  5.4  P r e d i c t e d e f f e c t i v e s t r e s s c o n t o u r s due t o b e n d i n g and b u l g i n g i n C a s e 1  67  5.5  Predicted principal b e n d i n g and b u l g i n g  s t r a i n v e c t o r s due t o i n Case 1(upper s h e l l ) .  68  Predicted principal b e n d i n g and b u l g i n g  s t r a i n v e c t o r s due t o i n Case 1 ( l o w e r s h e l l ) .  69  5.6 5.7  Relation between i n t e r n a l c r a c k s and c a s t i n g s p e e d a t O i t a No.4 c a s t e r *' 3  5.8  72  35  S u l f u r p r i n t of a l o n g i t u d i n a l s e c t i o n ; r a t i n g of i n t e r n a l c r a c k s = 0 . 2 "  74  S u l f u r p r i n t of a l o n g i t u d i n a l s e c t i o n ; r a t i n g o f i n t e r n a l c r a c k s =0.5"  75  S u l f u r p r i n t of a l o n g i t u d i n a l s e c t i o n ; r a t i n g o f i n t e r n a l c r a c k s = 1 . 0*  76  6  5.9  6  5.10  6  5.11 5.12  P r e d i c t e d b e n d i n g and b u l g i n g s t r a i n , Case 1 (upper shell,V=1.6m/min)  >in ...  e  P r e d i c t e d b e n d i n g and b u l g i n g s t r a i n , e , i n Case 1 ( l o w e r shell,V=1.6m/min) P r e d i c t e d bending  5.14  Case 2(upper shell,V=1.2m/min) P r e d i c t e d b e n d i n g and b u l g i n g s t r a i n , Case 3(upper shell,V=1.Om/min)  and b u l g i n g  strain,  P r e d i c t e d bending and b u l g i n g s t r a i n , Case 3 ( l o w e r shell,V=1.Om/min) Predicted distortion 1(lower s h e l l )  due t o b u l g i n g  78  E  x  5.16  77  x  5.13  5.15  .  11  n 79  £ x  , in 80  e x  , in 81 i n Case 82  X  Figure 5.17 5.18 5.19 5.20 5.21 5.22  Page P r e d i c t e d x x - s t r a i n due t o b u l g i n g (lower s h e l l ) '  i n Case 1  P r e d i c t e d x y - s t r a i n due t o b u l g i n g (lower s h e l l )  i n Case 1  83 84  P r e d i c t e d bending s t r a i n , e shell,V=1.6m/min)  i n Case  P r e d i c t e d bending s t r a i n , e shell,V=1.6m/min)  i n Case  Predicted curvature bending i n Case 1  1(upper 89 Hlower 90  of the s h e l l due t o 91  R e l a t i o n between bending strain, ,and roll p i t c h p r e d i c t e d by the f i n i t e - e l e m e n t bending analys i s  92  Relation between bending s t r a i n , e , a n d s h e l l thickness predicted by the finite-element bending a n a l y s i s ( M a c h i n e radius=10.5m)  92  Relation between bending s t r a i n , e , a n d s h e l l thickness predicted by the finite-element bending a n a l y s i s ( M a c h i n e radius=8.0m)  93  R e l a t i o n between bending s t r a i n , ,and machine radius(curvature) predicted by the f i n i t e - e l e m e n t bending a n a l y s i s  93  P r e d i c t e d t o t a l bending and b u l g i n g s t r a i n , j , a t an inner surface as a f u n c t i o n , of b u l g i n g s t r a i n , e , a n d bending s t r a i n , ^ ....  97  IV.1  Geometry of a t h i c k w a l l e d  115  IV.2  Comparison of the c a l c u l a t e d s t r e s s e s „(solid p o i n t s and lines) with those obtained by Hill ,  116  Predicted shell)  bending  119  Predicted shell)  bending  5.23  5.24  5.25  5.26  E  x  x  x  e  E  B  e  cylinder 0  3 0  VI.1 VI.2  strain, strain,  £  e x  , i n Case 2 ?  (upper  ,in  (upper  Case 3  120  xi  Figure VI.3 VI.4 VI.5 VI.6 VI.7 VI.8 VI.9 VI.10 VI.11 VI.12 VI.13 VI.14 VI.15 VI.16 VI.17 VI.18  Page P r e d i c t e d bending shell)  strain,e  of t h e s h e l l  b e n d i n g ' s t r a i n , e , i n Case  bending  P r e d i c t e d bending shell)  strain,E  bending  strain,e  5  (upper 125  strain,£ , i n 126  x  , i n Case 6  (upper 127  x  strain, .. e  strain,  ,in  e  128  x  , i n Case 7  (upper 129  x  bending  strain,  strain, ,in .. . E  x  , i n Case 8 .  e  bending  strain,  131  strain,  ,in  e  .  , i n Case  e  9  132  x  (upper 133  x  P r e d i c t e d b e n d i n g and b u l g i n g Case 9(upper s h e l l ) bending •  strain,  E  130  (upper  x  P r e d i c t e d bending and b u l g i n g Case 8 (upper s h e l l )  Predicted shell)  124  x  P r e d i c t e d b e n d i n g and b u l g i n g Case 7(upper s h e l l ) •  Predicted shell)  (upper  strain,z ,in  , i n Case .  P r e d i c t e d b e n d i n g and b u l g i n g Case 6(upper s h e l l )  Predicted shell)  4  123  P r e d i c t e d b e n d i n g and b u l g i n g Case 5(upper s h e l l )  Predicted shell)  due t o 122  P r e d i c t e d b e n d i n g and b u l g i n g Case 4(upper s h e l l ) Predicted shell)  (lower 121  Predicted curvature b e n d i n g i n Case 3 Predicted shell)  , i n Case 3  strain,  ,in  E  134  x  , i n Case .  10  (upper 135  x  P r e d i c t e d b e n d i n g and b u l g i n g Case I 0 ( u p p e r s h e l l )  strain,  e  ,in ?  136  xii  L I S T OF  d  slab thickness  d  grain size  0  (mm)  (y m)  [D]  material  E  Young's modulus  F  roll  i  [K] 1  Ri  SYMBOLS  matrix  friction  stiffness  (MPa)  force  (N)  matrix  rollpitch(mm) *  Al  elongation  (mm)  n  strain-hardening  N  number o f  exponent  rolls  necessary  shape  function  to a b s o r b the  bending  deformation [N]  matrix  p  ferrostatic  i  of  pressure  Q  self-diffusion  R  machine  R  gas  0  s  shell  T  temperature  T  "u u  x  Y  (J/mol)  (mm)  (J/mol°K)  thickness  surface  0  energy  radius  constant  (MPa)  (mm)  (°C)  temperature  average displacement displacement  (°C) in x direction  in y direction  V  c a s t i n g speed  w  slab width  Ay  distance  between o u t e r  y  distance  from the  (mm)  (mm)  (m/min)  (mm) and  inner  neutral axis  surfaces (mm)  (mm)  Zener-Hollomon parameter maximum b u l g i n g  (s~ ) 1  deflection  (mm)  strain strain  at inner  surface  strain  at outer  surface  strain  rate  (s~ ) 1  bending  strain  bulging  strain  total  of b e n d i n g and b u l g i n g  effective  plastic  curvature  of the s t r a n d  stress  peak Yield  strain  (MPa)  effective  stress  stress  frictional Poisson's  (MPa)  (MPa)  stress  (MPa)  coefficient ratio  strain  (1/mm)  xiv  ACKNOWLEDGEMENTS  I  wish  Dr.J.K.Brimacombe advice  to  express  and  Dr.I.V.Samarasekera  and g u i d a n c e t h r o u g h o u t Thanks  students  and f a c u l t y  Engineering. particular  The  also  Engineering i n Japan  must  also  be  the course  assistance  also  grateful  given  Finally,  for their  of t h i s  to  Council  of  the  my  to  useful  study.  fellow  graduate  to  technical  staff  and i n  i s greatly appreciated. to of  the  Natural  Canada  f o rproviding f i n a n c i a l  the plant  gratitude  members i n t h e D e p a r t m e n t o f M e t a l l u r g i c a l  Research  Corp.  sincere  extended  t h a t of Mr.N.Walker I am  providing  are  my  and  support.  Mr.H.Misumi  Sciences Nippon  Special  and Steel  thanks  for h i s assistance i n  data. I would l i k e  to  take  this  t h a n k my w i f e T e r u k o f o r h e r a s s i s t a n c e i n t h i s  opportunity work.  to  1  . 1  Chapter  INTRODUCTION  During  the  continuous-casting  last  two  machines  production  of  steel  vertical,  vertical  slabs. with  of  type  the l a t t e r ,  elongation,  can  solidification In  solidifying pressure pressure bulging  is strain  excessive  may  include  the  away In the  i s straightened  cracks,  bow  demands  has been  The r e s u l t a n t b e n d i n g g i v e s  while  rise  to  close.to  the  of the s t e e l  is  very  t y p e s o f s t r e s s e s a r e a p p l i e d to t h e  bulging  stresses  stresses. ' ' 1  relatively  the  a r c and o v a l  the trend  internal  of  for  i n Fig.1. With i n c r e a s i n g  cause  other  shell,viz.  and t h e r m a l  types  circular  f r o n t where t h e d u c t i l i t y  addition  developed  machine  however, the s t r a n d  core.  which  types  t o t h e low-head,'bow-type c a s t e r .  containing a liguid  low.  The  several  been  r a t e and p r o d u c t q u a l i t y  from t h e v e r t i c a l case  have  bending,  w h i c h a r e shown s c h e m a t i c a l l y on p r o d u c t i o n  decades  small  combine  2  1 6  due  to  ferrostatic  Even t h o u g h t h e  ferrostaic  i n a low head bow-type  with  bending  strains at the s o l i d i f i c a t i o n  strain  front.  This  to  caster, create  effect  will  vertical  type  vertical  type  with  Fig.l  bending  circular  arc  (one-point  Schematic drawing of D i f f e r e n t Types of Casting  type  bending)  Machines^?  oval  bow  (multi-point  type bending)  3  be shown i n t h i s As quality,  thesis.  increasing  emphasis  the  steels  since the l a t t e r  formation  of  ;  internal  half  s t u d i e s have been r e p o r t e d  cracks  on  and hence t h e t a s k  the  analysis  product  design  point  of  designs  a r e r e q u i r e d i n order  modern  low  view,  because  this  t o meet t h e  bending w i l l  step toward a b e t t e r understanding  be  from  a  machine  demands  for  work, the s t r e s s  considered  of bending  in  Further  subject  strict  cast  stresses  optimum and l i m i t i n g  head c a s t e r s . I n t h e p r e s e n t  of one-point  of  to  o n l y a few  i s by no means c o m p l e t e .  a r e u r g e n t l y r e q u i r e d on  analysis  on  in continuously  o f the 1970's. However,  investigations  the  been p l a c e d  s t r e s s a n a l y s i s h a s b e e n one o f t h e t o o l s e m p l o y e d  study  bending  has  as  behavior.  a  first  4  Chapter 2  PREVIOUS WORK AND O B J E C T I V E S OF PRESENT WORK  2.1  Internal cracks It  strains  face are  of internal cracks  solidification  bulging  t h a t even  steel  1  2  The  o r b e n d i n g o f s l a b s c a n be s e e n  formed  i n the upper s h e l l  cracks are generated  the  support since  residual  The reported  normal  i n upper and lower  strains  filled  for internal  t o be d e p e n d e n t o n s t e e l  low-carbon s t e e l s and low s t r a i n  bending  cracks  zone  shells  with  on  and  beneath sulphur  solute  rich  5.1.1). cracks  giving  reported values  due t o  t o t h e broad  grades and s t r a i n rates  to  have  been  rates  with  high  of c r i t i c a l  critical  strains.  Table  for  c o n d i t i o n s . Some s c a t t e r i n t h e d a t a ' ( 0 . 2 - 3.0%) c a n  these  presents  close  cracks  visible  ( s e e F i g s . 5 . 8 - 5.10 S e c t i o n  I  cracks)  i n the straightening  are generally  critical  tensile  can lead t o the  internal  p o i n t s . These c r a c k s a r e  they  liquid  shell  small  s e c t i o n . I n most c a s e s ,  bulging roll  slabs  (solidification  front." '"  i n a longitudinal  prints  cast  h a s l o n g been r e c o g n i z e d  applied t o the s o l i d i f y i n g  generation the  i n continuously  strain  5  Table  I  Studies  Study  of c r i t i c a l  strain  for internal  Method used t o o b t a i n critical strain  cracks  Critical strain  Ref  crit  Palmaers  B a s e d on e l a s t o - p l a s t i c t h e r m a l stress calculation of a c o n t i n u o u s l y cast bloom.  0.2%  1 2  0.39%  15  0.6%  36  0.3%  37  0.25-0.6%  38  2.0-3.0%  39  C=.18% e  Puhr i n g e r  ?  B a s e d on e l a s t o - p l a s t i c - c r e e p bulging calculation of a continuously cast slab. C=.05% e =6x10-« s" 1  Daniel  B a s e d on e l a s t o - p l a s t i c bulging c a l c u l a t i o n of a c o n t i n u o u s l y cast' bloom. C=.15% e  Nari ta  ?  R o l l m i s a l i g n m e n t t e s t on a c o n t i n u o u s l y c a s t bloom. S t r a i n i s c a l c u l a t e d based on e l a s t o - p l a s t i c s i m u l a t i o n model. C=.15% e =4x10-" s " 1  Suzuki  Reduction test . Low c a r b o n steel. E =1x10"* s " 1  Matsumiya  Laboratory 3-point bending t e s t . Specimen i sp a r t l y m e l t e d by e l e c t r i c a l input. C = .15% e = 5 x 1 0 - * s" 1  6  be  seen owing to the  d i f f e r e n t methods  critical  strain  difficult  t o measure the  value  has  a  value  adopted  i n the To  in I  ,  stresses  strand the so  as  are  steps are  and  machine.  2.2  Previous  this  point  the  internal steps  greatly  work on  some  authors has  been  b e f o r e the  whose a braking  reduced.*  3  p r e s e n t as  is  end,  bending  point  at  is  the  to rolls  electrically strand the  and  case  into  , of  several  solidification  bending c a s t e r s  type of  s t r e s s a n a l y s i s of  several  is divided  Multi-point a new  this  In  3  cancel  z o n e . To  to the  slippage. "  to  The  by  against  force  in  front  movement  straightening  to  bending".  method  bending  force  due  have been a d o p t e d  solidification  i n the  slab  cracks  "multi-point  hence t e n s i l e s t r e s s e s  under development a t bow  0.2-0.5% ;and  the  large  to apply  bending  with six  casting  required a  same t i m e , p r e v e n t  multi-point  i t is  .critical  the  two  at  force  with  bending  studies of  the  Since  l o c a t i o n , the  four  p r o b l e m of  compression  rolls  estimate  front.  these  range of  following  the  driven  controlled  front  this  to  study.  a compressive  following  bending  the  the  tensile  the  the  in  overcome the  of  applying  at  at  " c o m p r e s s i o n c a s t i n g " and  principle  push  strain  From T a b l e  present  straightening,  several  solidification  calculated  input.  report  practice:  the  been  experimental  harmful  at  used  low-head,  b e n d i n g and  are  oval-  bulging  7  As  mentioned p r e v i o u s l y , owing t o the  bending s t r a i n and very  easily  Bulging the  and  in  bulging the  in  1 2 1  , 5  Section  '  2 0  "  studies  2 7  ;  can  simultaneously  due  occur  to  study  to unbending.  have  been  to  bulging  the r e s u l t s w i l l be d i s c u s s e d  in  detail  3.3.2. However only a few 1 5  3 1  i n the l i t e r a t u r e  of c o n t i n u o u s l y  cracks  devoted  been r e p o r t e d on bending ' reported  analysed  of i n t e r n a l c r a c k s  Numerous  internal  of  s t r a i g h t e n i n g zone i n a bow-type c a s t e r .  bending must be  formation  analysis  strain  interaction  ' * *'"  and  5  mathematical models have no model has  on the b u l g i n g and  yet  been  bending a n a l y s i s  c a s t s l a b s which i s the s u b j e c t of the  present  thesis. The  s i n g l e beam theory,  at the c e n t e r  of the  explain  bending  the  slab thickness, behavior  According  t o t h i s beam theory,  one-point  and  are g i v e n  r e s p e c t i v e l y by the  bend  multi-point  E  ;  bend ;  U  long  been  continuously  the  bending  =  e  apparent from E q . ( l ) ,  with d e c r e a s i n g  - s) / R  I  cast  slabs. u  ,  for  in Fig.2,  .  O)  0  ) (I/R n  to  -I/R )  n-1  (2)  n  the bending s t r a i n becomes l a r g e r  s h e l l t h i c k n e s s and  assumption of one  s  used  strains, c  f o l l o w i n g equations  = (-—  u  is  of  has  m u l t i - p o i n t bending,shown s c h e m a t i c a l l y  one-point  As  which assumes a n e u t r a l a x i s  machine r a d i u s . However,the  neutral axis i s questionable  at the c e n t e r  of  Compression! (a)  one-point Fig. 2  bending  (b)  multi-point  bending  Elongation at the S o l i d i f i c a t i o n Front during Bending.  00  9  a wide face  of  a  slab.  analysis  Vaterlaus'  which the  u p p e r and  his  cracks  unbending and  the  distributions an  must a p p e a r  a s s u m p t i o n of  The  of  and  one  during  the  shell  present  formulated  To  as  theory  study  the  However  observations strains  i n the lower  by  Onishi  3 1  for  to shell  shell.  A  based  on  method.  calculation  used to e v a l u a t e  thickness  of  due  lower  the  because  it  in  of  strain  is  based  theory.  work has  been u n d e r t a k e n  casting conditions  s t a t e of  strain  partially  and  on  solidified  .  The  primary  to  elucidate  that crack  steel  elasto-plastic,finite-element  f o r t h i s purpose  a n a l y s i s are  (1)  on  unbending of  two-dimensional,  elastic  finite-element  suitable  neutral-axis  present  influence  in  c a n n o t be  m a c h i n e d e s i g n p a r a m e t e r s and strong  front  been r e p o r t e d  i s only  through the  t o the  tensile  elasto-plastic,  forces,  Objectives  harmful  solidification  t h i s model  reaction  neutral-axes  contradictory  m o d e l has  one-dimensional,  roll  2.3  are  , i . e . the  occur at  Unfortunately  three-dimensional  p r o p o s e d a two  hence i n t e r n a l c r a c k s  the  on  lower s h e l l s deform independently.  dynamic s i m u l a t i o n  on  has  4  model p r e d i c t i o n s  internal  Based  have  a  formation slabs.  model has  concerns  A  been  of  the  follows:'  determine the  whether  the  is correct  or  unbending b e h a v i o r , i . e .  conventional not.  to  ascertain  single neutral-axis  theory  10  (2)  To in  calculate unbending  reported (3)  To  the c r i t i c a l  find  i n the  and  to  strain  compare  for internal  cracks  i t  values  with  the  literature.  a correlation  between r e s u l t a n t t o t a l  strain,  e  , and e a c h of t h e c o m p o n e n t s o f b u l g i n g s t r a i n , , T B and b e n d i n g s t r a i n , E  e  u  The  present  the d e s i g n  one-point o f a new  bending  model w i l l  provide  low-head,bow-type c a s t e r .  the b a s i s  for  11  Chapter  BENDING/UNBENDING  3  STRESS A N A L Y S I S OF CONTINUOUSLY CAST S L A B S  3 . 1 Introduction Stress  analysis  has  been  performed  bending/unbending of p a r t i a l l y  .solidified  This  problem  is  through  a  very  the  alternately of and  straightening to  ferrostatic the  pushing  pushing  Thus each element At  the center  deformation  of  the  interaction  between  a  with a l i q u i d considered plastic  strand  wide  steel  slabs.  because  while  passing  strand  i s  subjected  solidified  the  shell  shell in  the opposite  of t h e s t r a n d e x h i b i t s plane is  of t h e wide enhanced  as  face  outward  a  complex  of the s l a b ,  a  result  b e n d i n g a n d b u l g i n g . On t h e o t h e r  deformation  formulating  the  the  against  hysteresis.  the c o r n e r ,  zone  the  t e n s i o n a n d c o m p r e s s i o n due t o t h e i n t e r a c t i o n  pressure  rolls  direction.  complicated  of  i s primarily  due t o b e n d i n g .  core,  bending  simultaneously  problem. This  problem and can r e s u l t  and b u l g i n g  deformations  hand a t  Thus,  model t o c a l c u l a t e bending o f t h e moving  of  in  strand  h a v e t o be  as a t h r e e - d i m e n s i o n a l , v i s c o - e l a s t i c -  introduces considerable in prohibitively  high  complexity  t o the  computing  costs.  12  In  order  to render the problem  following  to  the  shown t h a t is  be  a  shell  hollow  box  with  of t h e s i d e edge. Comparison  of  bulging analysis.  plane-stress  indicated  to the bending  formulated  2 1  analysis  f a c e of  Thus, a two-dimensional  formulation  been  of  the  i n c o r p o r a t e d but  of  i s related  strain  f o r b u l g i n g i f the r o l l can  be  c o n d i t i o n s are enhance instead  the  two  center can  i n the  case  has  been of  total  the  of creep  principal  to e l a s t o - p l a s t i c is  Since  but w i l l  deformation  sufficiently  that  cause  were  component  displacement  i t is anticipated  strain  i n the bending  2 2  elasto-plastic  the e f f e c t s  spacing  negligible.  imposed,  important  the  model  model  creep  small  boundary will  a stress  not  relaxation  analysis.  For the boundary c o n d i t i o n s , which most  of  with  s e c t i o n at the c e n t e r plane  straightening,  total  model  result  slab.  the  has  of t h i s  of w i d e s l a b s a s  not considered.. During  creep  gradients  t h a t a t w o - d i m e n s i o n a l model  f o r the l o n g i t u d i n a l  In  while  temperature  was  face  applied  the  was  the d e f o r m a t i o n at the c e n t e r p l a n e of the wide  be  behavior  analysis  have  of a s l a b has  t h e wide  elastic  of the wide s l a b , where t h e s t r a n d  of a t w o - d i m e n s i o n a l  plane  the  t h i c k n e s s . R e s u l t s of t h e c a l c u l a t i o n  independent  that  three-dimensional  the bending  considered to through  form  s t e p s were t a k e n . Firstly,  applied  i n t o a more t r a c t a b l e  aspect  factors specifically  of a mathematical  are  usually  model, the  h a v e been c o n s i d e r e d . The  the  following  calculation  13  has  been  performed  i n two  s t a g e s and  s u p p o r t s h a v e been a p p r o p r i a t e l y as  possible  a  moving  a  smoother  Secondly,  strain roll  approach  is  withdrawal bending  deformation  been u s e d as of  the  in that  this  force  i n the  in greater  elevated A  The  the  of  the  this  detail  i n the  of  low  on  i s the  accuracy  of  of  temperature  steel ,  data  solidifying utilized  conducted under c o n d i t i o n s  conducted to  interface. This  on  of  derived  from  the  friction  force  has  the  upstream  edge  a n a l y s i s . These t o p i c s  subsequent  the  are  sections.  steels  the  at  should  rate  be  few  years  determine  the  plastic  data. are  , thermal h i s t o r y , to  calculate  obtained in a  several  t e m p e r a t u r e s f o r a v a r i e t y of  study  temperatures  of c o n t i n u o u s l y  to those  past  property  elevated  In order  shell  similar  in a modelling  mechanical  strain  the  In  in  temperature  stresses  property  resulted  adopted concept  roll  carbon  chemical composition.  in  simply  considered.  is  boundary c o n d i t i o n  mechanical properties  elevated  force  slab. This  s t r u c t u r e and  the  been  usually  finite-element  and  roll  simulation  solid-liquid  f a c t o r of p a r a m o u n t i m p o r t a n c e  kind  dependent  the  has  from  as  semi-dynamic  lower s h e l l  at  force  Mechanical properties at  of  the  shell  discussed  3.2  friction  different  This  f o r the  distribution  resistance  chosen to simulate  strand.  proved u s e f u l p a r t i c u l a r l y  f o r each stage the  from  slabs tests  caster.  studies  behavior strain  cast  the  of  h a v e been steels  rates. " 3  1 6  at  Since  14  at  these temperatures  effects  of  obtained  for s t r a i n  continuous for  creep  temperature  It  of c r e e p  strain  this  range  difficult  to  elasto-plastic  range  Properties particularly  the  10"  important  the  to choose  data  encountered  strain  in  account  distribution  model. p r o p e r t i e s of s t e e l  solidus  from  out  to p a r t i a l l y  i n m o d e l l i n g the  study,mechanical  r a t e s i n the  those  i s thus p o s s i b l e  900°C t o  separate  i t i s important  r a t e s comparable to  s t r a n d , w i t h an For  is  from the data  casting.  the e f f e c t s  in the  it  temperature  t o 10~  5  2  s"  and  must be  1  i n the for  known.  are  1) Y o u n g ' s m o d u l u s , E 2) Y i e l d  stress,  3) P o i s s o n ' s 4) The  a  ^  ratio,  v  S t r a i n - h a r d e n i n g exponent,n  temperatures  and  values encountered  3.2.1  strain  i n continuous  T y p e s of s t r e s s - s t r a i n  Before proceeding is  important  to  at  stress-strain  1  are  typical  and  curves  property data, i t  t h e g e n e r a l f e a t u r e s of t h e  and  low  strain  rates.  Three  low  strain  rates,  1 1  ' * 1  The  flow stress  increases to a  peak  flow types iron  as shown i n  .  Type-1;  of  3  c u r v e s have been r e p o r t e d f o r a u s t e n i t i c  e l e v a t e d temperatures  Fig.3. 1"  casting.  to the mechanical  understand  curves at high temperatures of  r a t e s g i v e n above  value  15  C . % T'C(x100) 13 12 11 10 9 325 8 775 o A E=^7x1CJ / A • A 0.10 • x10" / A • • * *10~4 4  s  3  s  S  0.25  E=6.7><10"Vc; •  xia / 3  s  • x10" / e=5.7x10"4 2  S  4  0.39  * x10"/s 3  • x10' / 2  £=6.7*10/ 4  0.71  s  *  xia / 3  s  s  " *10"£  S  •TYPE-t  Fig.  3.1  ••• •• •• •••• ••• A  A  A  •  A  •  D  •  •  D  A  AO  ••• • • A  A  A  A  •  —  A  •  •  —  C  A  A •  •••  A  A  A  A  •  —  A  A  A  A  •  —  ATYPE-2  • • • • • •  • — —  aTYPE-3  S t r e s s - S t r a i n Curves f o r A u s t e n i t i c Iron at Elevated Temperatures and Low S t r a i n R a t e s . ^  16  and  then  about  Type-2;  a mean.  low  strain  The  flow  and ( Type-3;  falls  then  The  and  the the  initial true  3.2.2  strain  oscillates  temperatures  and  then  Type-1  and  increases  decreases  a  a  high  required  hardening  for  the  region,  level.  to  a  maximum  rapidly  strain  without  (  at  rates  )  present  that  value  3)  steady-state. and  peak  steady-state  of  stress  to  low  study  i s up  is  to  about  this  work.  for 1%  of  strain.  Mechanical  The 1)  data  to a  values  temperatures  stress-strain  high  which  stress increases  reaching  The  level  rates)  flow  value  ( at  falls  between  to a  property  f o l l o w i n g data  Young's modulus, The  data  from  obtained  as  tensile  independent  of  to  3x10"  1  as  follows.  by  tests  shown  tests  3  have  been  used  in  E  tensile  adopted  data  in show  strain  s" .-The  Mizukami and  for  7  a  a  0.08%C  resonance  Fig.3.2.  The  that  the  rates  i n the  dependence  of  E  method,  results  Young's region on  steel,  of  modulus from  were the is  1x10""  temperature  is  17  1000<T<1400°C E=1.96X10"-18.375(T-1000)  MPa ( 3 )  1400<T<1475 E=1 .225X1 0 " ( 1 4 7 5 - T ) / 7 5  MPa ( 4 )  E=0  MPa  T>1475  2) Yield  stress, The  Y  data  were  obtained  employed  strain has  „  rates  by N i e d e r m a y r  as  above  been o b s e r v e d  stress  (5)  shown 10"  2  in  s~ ,  The  5  ia  as  at  low  Fig.3.2. a  1  by J o n a s .  by N i e d e r m a y r  1 6  strain  rates  F o r the  higher  yield  stress  higher  formulation  of  yield  follows.  1000<T<1200°C o =66.15-4.655X10" T  MPa  2  y  (6)  1200<T<1480 MPa ( 7 )  O =54.39-3.675X10- T 2  Y  T>1480 a  3)  Poisson's  y  =  0  MPa  (8)  ratio,v  Poisson's dependent  ratio 1 7  '  1 8  ,  as  was shown  assumed in  v =8.23x10' T+0.278 5  to  be  temperature  Fig.3.2.  (9)  18  4 ) S t r a i n - h a r d e n i n g exponent, The f o l l o w i n g simulate  the  n  stress-strain  r e l a t i o n s h i p was used  plasticity.  a = K  E  n  (10)  where K i s a c o n s t a n t ,  E  is  the  the  s t r a i n - h a r d e n i n g exponent.  in  a  strain  to  t r u e s t r a i n , and n i s The exponent  n depends  complex way on such parameters as  temperature,  rate,  etc.  therefore  total  strain,  cannot  grain  be expressed  size,  by a simple  ,  and  equation.  However, a c o r r e l a t i o n has been observed r e c e n t l y Sakai  between the  6  first  peak  parameter,Z, size  stress  s t r a i n - h a r d e n i n g exponent a  under  _.  or  hence  conditions  of  Zener-Hollomon controlled grain  0  given as  Q  s-  o  where,  Q=(self-diffusion R=8.319 (gas  A,m are c o n s t a n t s .  increases » °  >or  energy)  J/mol°K  of  n i n t h e Y -phase  m o n o t o n i c a l l y with i n c r e a s i n g Z.  The  data  calculated  (11)  1  J/mol  constant)  The value  The  follows.  Z= £ exp(- R T  and  and the  (d =38 ^m and 42.3 P m ) as shown i n F i g . 3 . 3 .  Zener-Hollomon parameter i s  by  from  peak  stress  Palmaers , 1 2  I o  I  I  I  I  o  o  o  o  •  •  •  -»  N)  U)  • 4>  L o •  CJ1  Poisson's r a t i o , y  0.6  c  CD  c o  CL X  cu  cn c c  0.5  Sakai  h  6  0.036%C  d =42.3urn 0  S a k a i 0.16 % C 6  OA -  Palmaers  12  0.088%C  03  d =38umX  •—•'  0  ^—  TJ  o  0.2  c o  0.1  -4—»  LO  0  .1  10'  .  1  10'  .  i  10 8  i  1010  .  i  i  10 12  10  Sec -1  Q  Zener-Hollomon p a r a m e t e r , Z ( =£ exp (-5-7-) ) F i g . 3.3  Strain-Hardening Exponent as a Function of Zener-Hollomon Parameter,  o  0  0.1  0.2  0.3  0.4  Strain F i g . 3.4  0.5  0.6  0.7  0.8  (%)  Influence of Strain-Hardening Exponent on the S t r e s s - S t r a i n Curve.  0.9  1.0  22  although  the g r a i n  monotonic that  of  increase  n was  corresponding of  n  Fig.3.3 the  unknown,  n and  taken  but  from  some  this  resultant  shows  agreement  a  with  in.Fig.3.3. strain  F i g . 3 . 3 by  Zener-Hollomon  shows  also  i s i n good  f o r the present study, the  exponent  of  of  is  S a k a i , .as s h o w n  Thus  data  size  calculating  parameter  scatter,  hardening  of  Z.  The  about  curves,  plotted 0.05  i s u n i m p o r t a n t when v i e w e d stress-strain  the  as  in  in terms  shown  in  Fig.3.4.  3.3  Model  development Owing  problem major  to  certain  2)  Thus  in  is  the  has  been  moving  loading  used  problems.  the  bending/unbending  are  necessary.  The  follows.  narrow  face  is  neglected.  neglected.  with  well  are as  to the  of  of  the  EPIC-IV  stress/plane  condition  eminently  normal  program,  for plane  of  assumptions  here  formulation  finite-element Yamada  adopted  dimension  Creep  complexity  simplifying  assumptions  1) T h e  the  1 9  model, ,  strain  which and  some m o d i f i c a t i o n the  suited  slab. for  the two-dimensional was  developed  axisymmetric  to take  by  problems,  into  account  the  The  finite-element  method  is  solving  non-linear  and  complex  23  detail  in  3.3.1  The  validity  the  subsequent  Comparison  of  of  these  check  the  three-dimensional with  results  the  program,  ELAS65,  Group of bending  the  Assuming analysed. linear of  the  steel  the  The  was  temperature  was  The  excluded  three-dimensional Calculations  by  used  wide  slab.  Slab  2)  Shell  3)  Roll  4)  Bending  5)  Mechanical  a  considered  bulging  in  due  this  for  for  d  pitch  90 : 350  radius  :  10.5  properties:  model.  the  to  a  and  compared  A  computer  of  hollow  a  f e r r o s t a t i c pressure  mesh  the  schematic is  Analysis  shown  slab  box  through-thickness  A  view in  with  W  mm mm m  2  was a  direction of  molten of  the  Fig.3.5.  following-conditions.  l800 mm  a  three-dimensional  section  be  model,  Structural  half  the  to  : 250 x  thickness:  performed  Computer  finite-element  size  in  and  two-dimensional  was  analysis.  were p e r f o r m e d  1)  the  was  gradient  in  the  two-dimensional  symmetry,  slab  shell.  of  analysis  developed  of  examined  models  adequacy  Duke U n i v e r s i t y , analysis  three-dimensional  elastic  from  are  sections.  two-dimensional  To  assumptions  24  (outer) (middle)  T=1045°C, E=18767 M P a ,  v  =0.36  .... T = 1 2 3 5 ° C , E = 1 5 2 7 8 M P a ,  v  =0.37  (inner)  Owing  to  symmetry,  T=1425°C, E=8163  the  y-component  constrained  on  of  In the z-direction,  the slab.  roll  the longitudinal  supporting points  plane  perpendicular  bending  moments  downstream  section of  edge p l a n e  of the slab.  therefore  the  bend can  t r e n d may shell  was  left  Thus  about  be  and  the  their  strand  upstream  edge  loaded  with  deformation,  while  the  deformation  distribution  i t i s clear  be a t t r i b u t e d  to the  was  resultant  the strain  regarded  face  free.  the  shows  The  direction  after  was  of t h e wide  t h e nodes c o r r e s p o n d i n g  constrained.  planar  =0.39  v  displacement  center plane  the casting  shows  and F i g . 3 . 7  the slab  This  to  but kept  Fig.3.6 bending,  were  of  MPa,  that  individual  as independent  face  neutral  shells  axes  of the narrow  and face.  distribution  ratio(w/d)  to  i n the cross  t h e wide  to the temperature aspect  due  for  the  in case  analysed.  Fig.3.8  shows  analysis  obtained with  stress)  of  that  there  included,  the i s good  Therefore  model,  assuming  comparison  plane  agreement. would  of t h e wide  If plastic a c t even  stress,  i s  elastic  that  of  this  modeKplane  face. I t i s evident  behavior  more  i t c a n be c o n c l u d e d plane  of the results  the two-dimensional  center  the mid-face  edges.  a  were  independently the  to  be  of the  two-dimensional  sufficient  for  the  z  F i g . 3.5  Schematic Diagram of the Three-Dimensional Finite-Element Mesh for the Bending Analysis.  F i g . 3.6  Predicted Distortions of the Slab by the Three-Dimensional Bending Analysis.  Finite-Element  ON  27  B e n d i n g S t r a i n (%) c o  •H  m  1.0 0.5* A c o  •H  ra C Q) /  1  0.5 -1.0  //  j  CQ  ft E o o  ^  /  Fig.  3.7.  A  /  /  /  ^  '  /  '  ' ' /  Predicted XX-STRAIN D i s t r i b u t i o n i n the Cross Section of the Slab by the Three-Dimensional Finite-Element Bending A n a l y s i s .  28  29  bending/unbending  analysis  on  the center  plane  of the wide  face  of "the s l a b .  3.3.2  E f f e c t s of creep  In creep  in  the  the  calculations examining  following  bulging has  the  continuously  compared  slabs  their  Fig.3.9). plastic spacings  than  two m e t h o d s  data  bulging of  the  studies  included  a  this  Emi  cm)  in  bulging 1  and  and  '  1  5  that  creep  for  2  0  "  model with 2 0  (  2  7  and an see  the e l a s t o -  at' large but at small  small  the d i f f e r e n c e between  '  bulging  Sorimachi  of b u l g i n g  includes  2  obtained  comparison,  values  on  creep.  results  by  by  calculated  2 2  the  accomplished  finite-element  the  which  40  mm),  check  the v a l i d i t y  of  a n a l y s i s , a comparison  Wunnenberg  because Wunnenberg only  from  is  of  values  roll roll of  the r e s u l t s of the  is negligible.  To the  with  lower  the model than  5 <1 max  using  reported  predicts  spacings(less bulging(  creep  I t i s evident  model  have  neglecting  accuracy  This  Schwerdtfeger  results model  the  several  which  and  for primary  elasto-plastic  of  bulging  the e f f e c t of  on  evaluated.  results  cast  section  analysis  been  Grill accounting  i n c a l c u l a t i o n s of  data  of  2 7  and  the e l a s t o - p l a s t i c of model  Morita  made m e a s u r e m e n t s Morita  ,  shown  2 6  was  with  model f o r  predictions to considered.  large  i n Table  roll I I was  the  However spacings, used.  For  30  Roll  F i g . 3.9  Pitch  (cm)  Comparison of Maximum Bulging Predicted by the Creep Model and E l a s t o - P l a s t i c Model.  comparison was  purposes  selected  the finite-element  f o rthe elasto-plastic  Table  II  Kind  Measured  data  mesh  shown  i n F i g  analysis.  of b u l g i n g  by  Morita  2 6  Si-killed(40kg/ mm ) s t e e l g r a d e  of steel  2  Size Casting Shell  speed thickness  Surface Roll  temperature  pitch  Ferrostatic  i  m/min  55  mm  1000-1050  °C  399  mm  3*  Ferro-static Pressure u  1.1  i  i  i  .  j  j  .  2  mm  0 . 2-0.4  NODE 189 ELEMENT 320  i  mm  7.933 m(0. 54MPa)  head  Bulging  i  230x1230  u  n  n  u  i  Temperature U65"C E E  in  K t t t t K Iff  filtiWliPl*  1050 *C  199.5 mm F i g . 3.10  Schematic Diagram of the Two-Dimensional Finite-Element Mesh f o r the Bulging A n a l y s i s .  32  Mechanical  p r o p e r t i e s were  Figs.3.2,  and  rate  of 1x10"  stress  on  s"  3  strain  increased analysis stress  (Appendix  1  strain  assuming  plane  plane  calculations stress  The  condition  Strictly  which  many  has  degree  of r e s t r a i n t  the  c e n t e r of t h e wide  the  been was  face  small  s p a c i n g s ; . i n a modern  during  interruption  necessary  location  between  cm.  Under  of  casting  important  o f t h e maximum  rolls  that  good.  used  based  The  in  will  e x e r t s on t h e  plane  bulging  i n the present  condition  model  study.  depend  deformation  has been  b u l g i n g under  slab  on at  transient the  shown  t o be  c o n d i t i o n s of  caster roll  spacings  are  c o n d i t i o n s such  creep  to  the  downstream  causes  an  asymmetric  maximum d e f l e c t i o n  of creep  bulging shifts  s t r a n d movement a n d c r e e p  neglected  adopted  effect  between  the  than  The  model  will  as be  however.  Another the  plane  the elasto-plastic  commonly  forcalculating  30-40  the  i n Fig.3.11.  larger  between  elasto-plastic  accurate  approximately  strain  of the slab.  reasonably roll  of  in  become m o r e ' p r o n o u n c e d ' w i t h  of t h i s  t h e edge  shown  a uniform  results  i s usually  validity  the  data  bulging i s reasonably  authors  speaking,the  Thus  the  a r e shown  agreement  and t h e measured  by  I ) . The  and t h e d i f f e r e n c e s  bulging.  analysis  from  3 . 3 ( s e e S e c t i o n 3.2) a s s u m i n g  and plane  bulging  obtained  on b u l g i n g i s t h a t from  the  as a r e s u l t  deformation.  of 2  2  '  2  3  midpoint  interaction '  2  5  '  2  7  This  bulging deformation  but the influence  i s small.  this  i n the present  study.  Therefore  effect  was  on  also  33  0.9 E E  0.8  o Plane stress  0.7  A  0.6  PI ane st ra i n  | data  by Morita  26  LZ 0.5  cn 0.4 CD  0.3 0.2 0.1  0  0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8  F e r r o - s t a t i c P r e s s u r e (MPa) F i g . 3.11  ^  Comparison o f B u l g i n g S t r a i n s P r e d i c t e d by t h e P l a n e S t r e s s and P l a n e S t r a i n . F i n i t e - E l e m e n t A n a l y s i s .  0.0 6  c 0.0 5  2  0.0 4  I  o Plane stress A Plane strai n  0.03 cn <= 0.0 2 w  3D o.o 1h  CQ  0  10  20  30  40  N Number of divisions in a roll pitch  F i g . 3.12  I n f l u e n c e o f t h e Mesh S i z e o n B u l g i n g S t r a i n Elasto-Plastic Finite-Element Analysis.  i n the  34  In significant  a  influence  i n the c a s t i n g shown  in  finite-element  d i r e c t i o n h a s been e v a l u a t e d The  c a l c u l a t i o n s have  been  twenty  in  there  divisions  i s  little  p o i n t . Moreover  high  a  roll  computing  on  these  combined with  employed  for  pitch  this can  purpose  has  mesh  be  i n the e a r l i e r  regarded  this  as  study  will  since  result  approximated  by  interconnected  at a finite  in  assumptions  the  was c a r r i e d  out  i n subsequent  finite-element an  method  assemblage  equilibrium  points.  a  bending/unbending using  a  two-  of  a  sections.  Element  continuum  elements  generate  By  of  of  forces,  a s e t of l i n e a r l y  can  are  number o f j o i n t s o r n o d a l p o i n t s .  simultaneously  These  is  which  compatibility  d i s p l a c e m e n t s and the s t r e s s - s t r a i n law f o r t h e m a t e r i a l ,  nodal  this  s t r e s s ) , e l a s t o - p l a s t i c , f i n i t e - e l e m e n t model.  the  t h a t c a n be s o l v e d  a  cost.  bulging  In  to  Thus  r e s u l t s beyond  • 3.3.3 T w o - d i m e n s i o n a l e l a s t o - p l a s t i c F i n i t e  possible  a  size  evaluation.  of  divisions  o f the model a r e p r e s e n t e d  satisfying  size  and t h e r e s u l t s a r e  change i n t h e c a l c u l a t e d  dimensional (plane Details  utilized  t h e u s e o f more  Based analysis  conditions  f i n e mesh f o r t h e  prohibitively  mesh  on t h e r e s u l t s . The e f f e c t o f  Fig.3.12.  sufficiently  analysis,  then  for be  independent the  i t is  equations  displacements  used t o o b t a i n  at  the s t r e s s  35  strain  distribution  mathematical basis described  of  the  the  assemblage  of  elasto-plastic  elements.  The  finite-element  is  i n Appendix I I and I I I . The  present  in  computer program " E P I C - I V "  has been used i n t h e  1 9  s t u d y . The m a i n c h a r a c t e r i s t i c s o f t h i s p r o g r a m a r e :  1) Three-noded l i n e a r t r i a n g u l a r elements a r e used. 2)  The i t e r a t i v e  method  adopted t o solve 3)  the matrix  The i n c r e m e n t a l  Unloading  method  ( t a n g e n t modulus method)  iterative  material  Appendix I I I ) .  matrix  inversion procedure g r e a t l y  computing time f o r a n o n - l i n e a r this  technique  approaches  is  that  order  modifications  were  to  disadvantage  matrix  plastic  made.  use To  t h e main r o u t i n e  to  be  of  inverted  instability)  supporting  points)  this  program  facilitate  which c o n t r o l s  c h a n g e d . By t h i s m o d i f i c a t i o n (roll  the  The  reduces the  the  deteriorates.  In  simulation,  problem.  when  singularity(i.e.  convergence  non-linearity.  i s checked a t every stage of the  calculation(  The  i s assumed  plasticity.  i s adopted t o simulate 5)  inversion.  I s o t r o p i c hardening of the m a t e r i a l in  4)  ( C o n j u g a t e G r a d i e n t method) i s  a shift  was  made  the  the  following  semi-dynamic  the subroutines  i n the boundary possible.  was  condition  Secondly, sub  36  programs data  for plotting  processor.  the  The  results  following  element  mesh,  deformations,  contour  maps o f  s t r e s s e s and  To  check  stress/strain  pressure  solution  by  3.3.4  plane  this  available:  strain  post  finite-  vectors  and  accuracy in  a  of  the  thick-walled  program,the  cylinder  compared w i t h  the  under  analytical  Boundary c o n d i t i o n s  longitudinal  interest.  straightening of  are  a  strains.  computed and  been m o d e l l e d .  of  plots  as  Hill(Appendix IV).  The s l a b has  was  formulated  principal  the  distribution  internal  were  center  Fig.3.13  Three  and  z o n e were m o d e l l e d  domain were a n a l y s e d  plane  of  the  wide  shows a s c h e m a t i c one and  half the  separately  roll  view  a  the  in  lower  the  of  of  pitches  upper and using  face  the  shells  following  boundary c o n d i t i o n s . 1)  The  x-  n o d e s on values  and  y-ccmponents of  t h e d o w n s t r e a m e d g e were s e t of  geometrical  beam b e n d i n g 2)  The  displacements  were  displacement  constrained.  based  to on  the the  simple  CD  since  s t r a i g h t e n i n g i s a s s u m e d t o be roll  were  pitch  also  roll  supporting  y-component of  AB  first  and  The  of  along  the  equal  of  theory.  y-component o f  points  the d i s p l a c e m e n t s  nodes  constrained(Fig.3.13),  downstream  completed from  the  within bending  Fig.  3.13  Schematic Diagram o f the Boundary F i n i t e - E l e m e n t Bending A n a l y s i s .  Conditions Adopted  i n the Two-Dimensional  38  point. 3)  The  3 1  roll  friction  deformation adjacent  The  were  caused  uniformly  rolls.(Coefficient  assumed 4)  forces  by  bending  distributed  between  of  roll  friction  was  t o be 0.45)  upstream  through  the  equivalent  edge  of  force  the  shell  boundary  was  constrained  condition;  t o the constraint force  from  this  the  i s  remaining  doma i n . 5)  The  roll  analysis  points and  were  the  shifted  above  once  boundary  i n the bending  conditions  were  reapplied.  3.3.4.1  Roll  The direction see  does  Fig.3.6  the  face narrow  of  Section  were  free  of t h e upper face  lower  shell  plane  t o move  frictional  cross  not remain  consideration wide  friction  during  would  section perpendicular planar  after  3.3.1).  If  would  unbending,  between  the  the  deformation(  section  the center  plane  move d o w n s t r e a m while  move u p s t r e a m . to  to the casting  the bending  a t the ends,  shell  relative  forces  force  This  narrow  the r o l l s  face  plane  to  of the  of the center  i s  and the s t r a n d  of the  relative  the center tendency  under  opposed  by  s u r f a c e , see  Fig.3.13.  The to  roll  the following  friction  steps:  f o r c e h a s been e s t i m a t e d  according  39  1) C a l c u l a t e edge  movement  of t h e downstream  /TT .  2) A s s u m e the  the average  t h e number  above  of r o l l s , N ,  necessary  to  absorb  d i s p l a c e m e n t ,~u . x  N  3) C a l c u l a t e  the cumulative  roll  friction  forces,  V  F. =  where  p  : roll  Ri  (12)  pressure  pitch  : frictional  y  4) C o n v e r t  Ri  : ferrostatic  i  1  X  to stress  E  %F . ±  i=l  coefficient °  a t each  i  (=0.45  roll  )  point.  =( £ F / S h e l l ) x 4  c  (13)  i  I  (stress  i s the total  CT i  different 5) D e c i d e from  of four l a y e r s  of  materials.)  strain  corresponding to the stress i the s t r e s s - s t r a i n curve,  6) C a l c u l a t e  elongation  A  i  1  =  R1 '  X  7). R e p e a t p r o c e d u r e s convergence  E  A l from  i *  from  1  0  strain  2  2) u n t i l  a.  (  1  4  )  final  i s achieved.\  N 1  i=l  h 1  = \ x  (15)  40  Appendix  V  presents  see  T a b l e III  and  N is  of  Section 4),  this  c a l c u l a t i o n of C a s e 1(  where the v a l u e of  strongly  value  affects  Fig.3.14.  the  Case 2(  the  of  the  coefficient  resultant  bending  calculation conditions  see  Table III  of  Section  4).  continuous casting  adopted  empirically  for  in  the d e s i g n .  The v a l u e of  0.33  of d r i v i n g  rolls;  Fig.3.15.  The  temperature;  friction  reported  to  friction.  The v a l u e of  be  corresponding the  is  and t h e r e f o r e  to  present  casters  about 9 0 0 ° C t o  one  the  of  seen  the the  iron  rolling  to d e c r e a s e oxide  main see  analysis,  since  Fig.3.15,  3 . 3 . 4 . 2 S h i f t of  i n the  bending p o i n t .  is  friction has  been  however  a  margin  therefore,  with  increasing  thickness which  see  is  affects  Pavlov  in  air)  9 5 0 ° C was a d o p t e d  one-point the  as  used,  variables  temperature of  1 0 0 0 ° C at the  in  was  film  s u r f a c e temperature of  surface  To take  for hot  0.45(  analysis,  the  to provide  In the p r e s e n t 3 2  shown  roll  v a l u e appears to be an u n d e r e s t i m a t e safety  as  of  the d e s i g n  the d a t a summarized by S c h e y  type  slabs.  friction  Unfortunately there  of  for  mm  f o r which a r e the same  for  of  8.86  roll  strain  f o r the c o e f f i c i e n t  this  x  of  no measured d a t a a v a i l a b l e the  is  u  11.  The  in  an example of  bending  bow-  s t r a n d ranges  from  1 2  the boundary c o n d i t i o n  i n t o account  the  effects  of  a  moving  41  B.i P i  Roll  Fig.  3.14  -1  i  0  x  i  1  No..at s t r a i g h t e n i n g zone  Influence of C o e f f i c i e n t of R o l l F r i c t i o n on the Resultant Bending S t r a i n .  42  Temperature ( ° C ) Fig.  3.15  C o e f f i c i e n t of R o l l F r i c t i o n of Hot R o l l i n g as a Function 3?  Temperature.  J  43  Fig.  3.16  Predicted Bending Strain with the One-Step Bending Model.  44  Roll  -10"  No. at straightening zone  inner surface  C  D  outer surface  c c  CD  CD  -1.5 -  • CASE 1  -2.0.1 F i g . 3.17  1  5 '  L.  Predicted Bending and Bulging Strain with the One-Step Bending Model.  45  strand,  the  complete  dynamic  by  steps;  small  computing  roll  points  were  however  this  results in  s h i f t e d only  once d u r i n g  bent  in two-steps  around  results  of  bending,  A  especially  in  bending  are  smoother  calculation, were  with  strain  peak  bulging  ,see  cause  significant  a  3.3.5  steps,i.e. Fig.3.18. by  a half  where In  to of  the  two-step  in  "Bending"  of  the  the  see  of  shell.  and  i n the  Compare  note  outer  that surface  calculation  one-step  inner  of  single-step  and  the  r e s u l t s of  Fig.4.9  observed  bending  surface  i n t e r a c t i o n between  i n the  3.19 and  separately.  After  was  which  bending  t h e s e peak one-step  and  strains  bending.  flow  and  first-step  strand  bending  Figs.3.16,3.17 . I t i s believed error  the  5)  inner  high have  lower  (Chapter  the case  points  points  points( was  a  roll  those  i n the a  a r e s u l t of  Figs.3.18  calculated  case  distribution  Calculation  calculation.  compared  roll  For  prohibitively  hence  difference  s t r a i n s appear as  and  bending  Figs.5.12,5.20  employed.  magnified  two  the  f o r the case  was  bending  significant bending  the  procedure  the  two-step  Figs.3.16,3.17  a  In the p r e s e n t a n a l y s i s , the  been  4).  bending.  modelling i t i s necessary to shift  cost.  Section  shifted during  the  "bending  Each  bending  first-step  p i t c h t o s e t new  shows  flow  plus  calculation and  and  f o r the  bulging" consists  second-step  bending,roll  boundary  chart  points  loading  were of  two  bending,  see  were s h i f t e d  conditions  for  46  the  second-step bending a n a l y s i s .  points  were  unloaded to allow  the  roll  p o i n t s were l o a d e d  new  push roll  the  strand  back  The n o d e s o f t h e f o r m e r  roll  f o r a s p r i n g b a c k . The n o d e s o f with  the  to simulate r o l l  displacements,  U  / Y  , to  c o n s t r a i n t a t t h e new  points. In  ferrostatic stage.  the  calculation  pressure  was  of  loaded  "bending in  plus  bulging",  the second-step  bending  47  c  READ INPUT DATA  3  GENERATE MESH TOPOLOGY NODE COORDINATES  c  PRINT INPUT DATA  SET NEW BOUNDARY & LOADING CONDITIONS 'ELASTO-PLASTIC ROUTINE"  c  I  |  APPLY FERRO-STATIC J  J  PRESSURE  j  PRINT RESULTS •ELASTO-PLASTIC ROUTINE"  MOVE ROLLER POINTS BY A HALF PITCH  c F i g . 3.18  PRINT/PLOT RESULTS  3  Flow Chart f o r the C a l c u l a t i o n of the Bending and Bulging Strain.  48  1 START  ELEMENT  COUNT  N=1 ,2  NE  COMPUTE  ELEMENT  STIFFNESS  CALCULATE  INCREMENTS  OF  AND  STRAIN  THE  MINIMUM  STRESS  CALCULATE  MATRIX  LOAD  RATIO(R  M  |  (  (  )  FOR  ELEMENTS ASSEMBLE  MASTER  STIFFNESS  MATRIX DO S^.  ANY  RECOVER  ELEMENTS ELASTICALLY7^  YES  MIX  NO  MIM  YES  SOLVE BY  EOUATIONS  C . G .  METHOD  ADD  THE  INCREMENTS  THE  PREVIOUS  TO  VALUES  V  19  Flow Chart for the C a l c u l a t i o n of the Bending and Bulging Strain. ( " E l a s t o - P l a s t i c Routine" >  49  Chapter  4  CALCULATION CONDITIONS  Calculations  have  bending bow-type c a s t e r s understanding Multi  been p e r f o r m e d o n l y  i n an a t t e m p t t o o b t a i n a  of  such  machines should allow  due t o c r e e p t h u s m a k i n g c r e e p parameters as  that  1) M a c h i n e R a d i u s , R Roll  Pitch, 1  3) C a s t i n g 4) S h e l l  analysis  were i n v e s t i g a t e d  follows;  2)  fundamental  of bending/unbending of c o n t i n u o u s l y  b e n d i n g b o w - t y p e c a s t e r s h a v e n o t been  analysis  f o r one-point  R  Speed, V  Thickness, s  5) Surface Temperature, T  Q  6) F e r r o s t a t i c  p  Pressure,  cast  slabs.  considered  since  for stress mandatory.  relaxation The  main  w i t h the computer model are  50  However  the three  thickness  and  variables; casting upon  surface  the latter  speed w h i l s t machine  temperature works(  have  The  in  shell  obtained  from  the  at  Oita  chosen  as  a  and  data  shell  and  dependent  and  surface at  Oita  has  been  thickness  and  distance/casting  conditions  works of Nippon S t e e l  base  case  for  t o b e made b e t w e e n  on  cracks  specifications  only  (= a x i a l  operating  comparison internal  is  radius  temperature  of  time  machine  plant  surface  against  independent  thickness  the  A plot  not  shell  Fig.4.1.  design  caster  are  pressure  The  where  temperature  pressure,  a r e d e p e n d e n t upon  for simplification.  i s shown  ferrostatic  ferrostatic  been  6  mid-face  two  -  temperature  radius.  NSC)* ,  smoothened  speed)  parameters  the  Oita  model  No.4  Corporation,  calculation  to  unbending.  caster  are as  the NSC  p r e d i c t i o n s and  r e s u l t i n g from the  of  of  3 5  ,  slab were  enable plant  The  follows(  data  machine see  also  F ig . 4 . 2 );  I)  machine  II)  roll  III)  slab  IV)  chemical  radius  : 10.5  m  pitch  : 47 1  mm  size  : 250 x d  composition  of  (Al-Si-killed(40kg/mm ) 2  1300  -  l900 mm W  slab steel  grade):  0.15/0.19%C,0.lO/0.30%Si, 0.70/0.90%Mn,0.025%>P,O.Ol5%^S,O.Ol/0.04%T,Al V)  threshold  casting  speed f o r bending 1.1  -  1.2  m/min.  internal  a  cracks:  o  1500  ""'HOO  0  2  4  6  8  10  Time below g. 4.1  12  14  16  18 •  20  meniscus ( min.)  Surface Temperature and Shell Thickness i n the Continuous Casting of Slab.  22  1.  B.  Ferro-static Pressure (M  a  O  3  Roll Pi tch F i g . 4.2  R o l l P r o f i l e of the 10.5  Radius Caster,  ( mm )  Table III  Calculation conditions for unbending of continuously cast slabs.  Machine Radius R  m  Roll Pitch HR  Casting Speed v m/min  Ferrostatic Surface Shell Pressure Temp. Thickness P MPa To ' C S mm  10.5  471  1.6  83  930  0.74  2. U  10.5  471  1.2 *  97  900  0.74  3. U,L  10.5  471  1.0  106  900  0.74  4. U  10.5  471  1.2  97  990  0.74  5. U .  10.5  471  1.2  97  850  0.74  6. U  10.5  400  1.2  97  900  0.74  7. U  10.5  540  1.2  97  900  0.74  CASE 1. U,L  8. U  8.0  471  0.9  97  900  0.56  9. U  8.0  471  1.2  83  900  0.56  10. U  13.0  471-  1.47  97  900  0.91  TJ : upper s h e l l ,  L : lower s h e l l  * : threshold casting speed f o r i n t e r n a l cracks  to  Ferro-static Pressure (MPa) LT  ir d  ID  in d  cr cr LT LT  LT)  CT LT  in  <3 a  tTTTTTTTTTTJ a  Roll Pitch (mm) R o l l P r o f i l e of the 8.0 m Radius Caster.  E=L\  t  B. P  Ferro-static Pressure (MPa)  Roll Pitch  F i g . 4.4  R o l l P r o f i l e of the  Radius Caster.  (mm)  56  The conditions which were investigated in the present study are given in Table I I I . The slab thickness was mm  for  a l l cases.  In Cases 1,2 and 3, the casting speed  varied to evaluate the c r i t i c a l of  kept constant at 250  s t r a i n necessary f o r generation  i n t e r n a l cracks. In Cases 4 and 5, the  was  artificially  surface  temperature  changed using the same conditions as in Case  2; and in Cases 6 and 7, the r o l l p i t c h was changed. 8,9  and  10,  machine  In  Cases  r a d i i of 8 m and 13 m were studied; the  r o l l configuration for these hypothetical machines i s shown Figs.4.3 and  was  in  4.4.  The  temperature  distribution  through  the  shell shell  thickness has been assumed to be l i n e a r with  the  inner  surface  of  Al-Si-killed  at  the  (40kg/mm )steel  grade  2  distributions  -  temperature  1487°C .  Based  separately  with a heat-flow model ,  33  calculated  t h i s assumption thickness  solidus  temperature  i s very reasonable. The temperature  were  and  r o l l pitches  the model. This again i s a very reasonable mechanical  being  considered  shown  in  Figs.3.2  properties for the above  and  3.3(  see  temperature  Section  d i f f e r e n t properties were assigned,to each of the into  which  the  shell  was  divided.  by  approximation.  d i s t r i b u t i o n have been c a l c u l a t e d from the mechanical data  shell  assumed to be uniform in the casting d i r e c t i o n  over the three-and-one-half  The  on  property  3.2.2) four  To determine  and  layers  the s t r a i n  hardening exponent,n,the s t r a i n rate in each of the layers  was  57  calculated  by  considering  halfway  through  strain  rates  approximation scatter strain  curves  Fig.4.9.  number  must  be  thickness. used  i s sufficient  a s shown  The in  the slab  the neutral  to  since  Strictly determine  the data  i n Fig.3.3. Figs.4.5  c a l c u l a t e d by t h i s  finite-element The  of elements  total  mesh  number  equalled  plane  of bending  speaking n,  of n  t o 4.8  the  real  however  this  itself  show  t o be  the  has  some  stress-  procedure.  for this  c a l c u l a t i o n i s shown  o f n o d e s a m o u n t e d t o 536 a n d t h e  924.  _  50  0.1  0.2  0.3  OA  Strain F i g . 4.5  0.5  0.6  0.7  0.8  0.9  1.0  ( %)  Assumed Stress-Strain Curves for the Slab i n Case 1.  oo  60 No.  T(°C)  6 ( V )  1  1413  10~  2  1266  3  1120  10 10"  4  974  10'  50  £  40  E Z  S  5  -4  tT=6.0£  a11  (T=22.5£  alA  A  0*=62.7£  A  CJzlOlOE 0  a21  23  30  0.3  0.4 S t r a i n  Fig.  4.6  0.5 (%  0.6  0.7  0.8  )  Assumed S t r e s s - S t r a i n C u r v e s f o r t h e S l a b i n C a s e 2,3,6,7,8,9 and 1 0 .  0.9  1.0  50  40  -  No.  T (°C)  6 ( >s)  1  1424  10~  5  2  1300  10~  4  CJ=17.6E  3  1176  10"  4  CJ =48.06  4  1052  10"  A  0"=85.2E  0" = 5 . 1 £  Q11  ai3  a2  Q23  Strain F i g . 4.7  ( % )  Assumed Stress-Strain Curves f o r the Slab i n Case 4.  o  0.2  0.3  0.4  Strai n Fi  8. 4.8  0.5  0.6  0.7  ( % )  Assumed Stress-Strain Curves for the Slab i n Case 5,  0.8  0.9  1.0  NODE  536  ELEMENT  924  \  /'  1  STAGE  UNBENDING  STAGE  UNBENDING  F e r r o - s t a t i c pressure —>  X  F i g . 4.9.  Schematic Diagram of the Two-Dimensional and Bulging Analysis.  Finite-Element Mesh f o r the Bending to  63  Chapter  MODEL P R E D I C T I O N S  5.1  Results  plots  analysis  from  x  component  the  support  and  of  inner points  a  shell  the  value  (about  0.1%),  a s shown  cracks  could  strain,  the while  upper  The d e f o r m a t i o n  e x  shell  t o appear  seen  strain  Thus,  the  and w i l l  - 0.65%)  be  occur  beneath  the  roll  surface  of  the  strain,E  This  As  cracks.  (0.55  at the inner  i n Fig.5.6.  i n the s t r a i g h t e n i n g  vectors.  strain  to internal  of t h e peak  be e x p e c t e d  contours,  of p r i n c i p a l  principal  reference  i n Fig.5.5,  strain  and  of the computer  o f XX- a n d Y Y - s t r a i n .  i s  of  bending  XY-strain  directions  of t e n s i l e  surface  of  i n terms  contours,  the  as those  with  results  and p r i n c i p a l  strain  peaks  the  XX-strain  lower  shell  DISCUSSION  1,presented  5.6,  hereafter  High at  f o r Case  a r e t h e same  discussed  show  stress contours  Figs.5.5  vectors c  t o 5.6  of deformation,  effective  AND  of c a l c u l a t i o n s  Figs.5.1 bulging  5  implies  i s rather that  preferentially  on  small  internal the upper  zone.  of t h e upper  and l o w e r  shells  i s  not  F i g . 5.1  Predicted D i s t o r t i o n Due to Bending and Bulging i n Case 1.  XX STRAIN  iB.P. 1 2 3  -0.3944 0.1036 0.2112 n 0.3188 5 0.4264 0.5340 6 7 0.6416 8 = 0.7491 0.8567 9 0.9643 10 ES  ZS  «•>•  S3  CS  E-•04 E-•02 E--02 E--02 E--02 E--02 E--02 E--02 E--02 E--02  XX STRAIN 1 2 3 4 5 6 7 8 9 10  F i g . 5.2  _ -0.1273 E - 01 as -0.1120 E-•01 = -0.9666 E- •02 -0.8132 E--02 = -0.6598 E--02 as -0.5064 E--02 -0.3530 E--02 « -0.1996 E--02 ea -0.4625 E--03 et 0.1071 E--02  Predicted XX-STRAIN Contours Due to Bending and Bulging i n Case 1.  ON  XY STRAIN  1  = -0.1009  E--02  2  -  -0.6552  -  -0.3016 0.5195  E--03 E--03 E--04  S3  -  0.4055  E--03  0.7591  E--03  7 =  0.1113  E--02  8  0.1466 0.1820  E--02 E--02  0.2174  E--02  3 4 5 6  9 = 10  S3  XY I  -0.3477  E-02  2  -0.2940  E-02  3  -0.2402  E-02  4  -0.1865  E-02  5  -0.1327  E-02  6  -0.7895  E-03  7  -0.2519  E-03  8  0.2856  E-03  9  0.8231 E-03 0.1361 E-02  10  F i g . 5.3  STRAIN  Predicted XY-STRAIN Contours Due to Bending and Bulging i n Case 1.  EFFECTIVE STRESS (MPa)  i\ i LT i  I  1 2 3 4 5 6 7 8 9 10  0.4806 0.7817 0.1082 0.1383 0.1684 0.1986 0.2287 0.2588 0.2889 0.3190  E+01 E+01 E+02 E+02 E+02 E+02 E+02 E+02 E+02 E+02  ^  EFFECTIVE STRESS (MPa) 1 2 3 4 5 6 7 8 9 10  F i g . 5.4  = = = = = = = = =  0 .5478 0 .8607 0 .1174 0 .1486 0 .1799 0 .2111 0 .2425 0 .2738 0 .3050 0 .3364  E+01 E+01 E+02 E+02 E+02 E+02 E+02 E+02 E+02 E+02  Predicted EFFECTIVE STRESS Contours Due to Bending and Bulging i n Case 1.  ON  1.0 %  Tension  J3. P/  i  r  i,.  llsli  — —  Compression  i-  4  i  — | — |— +- -.. — | — .— -> —  — _ —  —  -  1  • — *— i —  _ 1*— —i- —• — . —  1  ,K  v  1  F i g . 5.5  J  _ -=  :L  I" -r "I - I " F  U  1  I  I  'J  -  J_  -  |—  ^-4—  -  -  ~  _ I_  1_  _L  I  T  [  "  1i. - Ii ^  ~*  I  J  _  1  _  L  1  J  .1  I  Predicted P r i n c i p a l Strain Vectors Due to Bending and Bulging i n Case 1.(Upper Shell)  oo  1.0  ^  —  r i  -41  -t  Fig.5.6  1  ^  _  R  Iv-  l**- yi  L  1  j  -  r-  -rV  7V-  H"  -fV-  -vV-  I'-  V-  -V-  4^^- -ty  >-  -V  -I — i -  — i — ,— —  -V-  r  —  I  I.I  1  Lv- J**-V  —i-  .  Compression Tension  1  U"  i  %  f  I  +  "I  -/V  r _ r _ i _  I I  I  I  —  ~  —  i_ .t _,  _,_.  -  -rV  — — — -«—i 1  —  —  I  -Ar -A*- -!v- -V"  - ] 4 / — — f V - -fV—fV—jV  i  I  —  1  h  -V  -  1-4-  Predicted P r i n c i p a l S t r a i n Vectors Due to Bending and Bulging i n Case 1.(Lower Shell)  ON  Tnblc IV Strains at s o l i d i f i c a t i o n Bulging  e * X  CASE 1.  front on the center plana normal to the wide face.  Ber iding  Bending and B u l g i n g  inner outer surface s u r f a c e e *  U  0.08  L  0.08  2.  U  0.062  0.1  0.85  3.  U  0.045  0  'L  0.045  4.  U  5.  0.17  *  X  X  E  0.84  z  *  E  e" P  xy  %  0.55/0.65  -0.26/-0.28  -0.27/-0.33  0.043/0.16  3.1/3.4  0.5/0.67  0.1  -0.0075  -0.085  0.09  3.8  0.56  0.25/0.3 * -0.13/-0.18  -0.12/-0.11  0.043/0.06  3.2/3.7  0.23/0.29  0.84  0.15/0.2  -0.078/-0.12  -0.08/-0.1  0.033/0.016  -0.02  3.2/3.3  0.14/0.2  -0.85  0.1/0.13  -0.037/-0.09  -0.086/-0.04  0.065/0.055  3.5/3.1  0.28/0.16  0.072  0.15  0.91  0.38/0.4  -0.19/-0.15  -0.13/-0.17  0.0058/0.093  U  2.8/2.9  0.3/0.32  0.055  0.03  0.79  0.15/0.23  -0.077/-0.15  -0.076/-0.09  0.011/-0.001  U  2.4/4.0  6.  0.034  0.09  0.15/0.20  0.14/0.34  0.84  -0.06/-0.12  -0.08/-0.09  0/0.012  U  3.5/3.3  0.17/0.18  7.  0.089  0.1  0.87  0.3/0.4  -0.11/-0.25  -0.12/-0.17  0.055/0.058  3.4/3.6  8.  U  0.039  0.27/0.42  0.09  1.1  0.15/0.2  -0.086/-0.098  -0.077/-0.10  0.0075/0.004  2.9/3.6  9.  U  0.063  0.19  0.22/0.17  1.05  0.35/0.4  -0.19/-0.15  -0.17/-0.084  0.007/-0.04  3,3/3.6  10.  u  0.082  0.11  0.33/0.27  0.73  0.3/0.33  -0.11/-0.16  -0.13/-0.12  0.056/0.0057  3.2/3.5  0.24/0.26  -0.21  -0.82  U : upper s h e l l •»  ,  L : lower  critical strain for internal  shell cracks  71  symmetrical, the  upper  for is  as  shell  example 0.87  mm  shown  Maximum  is usually larger  the  and  in Fig.5.1.  than  maximum b u l g i n g  0.39  mm  of  bulging deflection  that  the  respectively  of  of  the  lower  shell,  upper  and  lower  shells  between  No.l  and  No.2  rolls.  As each  contour  that  the  seen  from  the  i s uniform  longitudinal  contours,  E  through strain  the  Fig.5.2,  shell  the  spacing  thickness,  distribution  indicating  is linear  £  of  through  X  the  shell  similar  thickness. to  dimensional close  to  one  that  2 to  5.1.1  the  IV  rating crack  of  case 5.10  of 0  6  through  speed  other  model  slabs  with  of  shell  rather  than  of  e  NO.4  sulfur internal  see  fairly  The  of  with  crack  to  prints crack  be  1.1  plant  a  results  data  internal 3 5  cracks  Here,  length divided  1.2  compression. of  the  VI.  speed -  at  remaining  3  casting  two-  small,  strain  caster, NSC. "' as  a  is  Fig.5.3.  between  i s defined  i s seen  is  i n Appendix  ordinary casting i . e . without show e x a m p l e s  each  cases.  prediction  threshold  cracks  of  components  relation  Oita  cracks the  strain  presented  at  Thus  internal  are  beam  a l l the  for the  of  behavior  component,  shows t h e  internal  spacing.  related  10  Comparison  casting  shear  lists  front  Case  the  simple  f o u r t h of  Fig.5.7 and  a The  solidification Case  of  continuum.  Table  of  Thus,  for m/min  r a t i n g s of  by  bending for  Figs.5.8  longitudinal  the  the to  section  0.2,0.5 a n d  1.0  72  respectively.  The i n t e r n a l  segregation apparent the  lines  from  upper  E  generally  between, t h e p r i m a r y  these  shell  cracks  with  prints,  internal  an increase  appear  as  arms o f d e n d r i t e s .  cracks  tend  i nc a s t i n g  to  dark As  appear  i s on  speed.  1.0  r 0.8  1.0  1.2  1.4  16  1.8  Cuiing «peed (tn/min)  Q Ordinary casting,  Fig.5.7  Relation between Internal Cracks and Casting Speed at Oita NO.4 caster, C Al-Si-killed  Figs.5.11 casting  Q Compression casting  (AOkg/mm ) s t e e l grade)  to  5.15 show t h e m o d e l  predictionsf o r  s p e e d s o f 1 . 6 , 1 . 2 a n d 1.0 m / m i n . T h e l e v e l  of strain  e  x at  the  increase lower  inner  shell  0.30%  of  i nc a s t i n g speed,  cracks,which to  surface  remains  according  critical  strain  values  reported  while  l o w . Thus  i sreached to  e  shell  t h ec r i t i c a l  the present  the  increases  a t the inner  a t a c a s t i n g speed  i sresonable i n  t h e upper  surface  strain  o f 1.2m/min,  analysis.  i ncomparison literature  for  t o the (refer  This  w i t h an of the internal i s 0.25 value of  experimental to  Section  73  2l)12»15t36-39  5.1.2  Bulging  strain  Figs.5.16 analysis tensile  of  Case  strain,  e x  of  the  shell.  one  half  of t h e  fairly  5.18  to  1(lower ,  shell).  appears  in  component  x  important  due  with  t o b u l g i n g by  presented  in  from  beyond which lower  shells  The t o t a l  is  f i n d i n g has  deflection  and b u l g i n g i s u s u a l l y  larger  between bulging  due t o t h e than  that  itself.  and  strain,  E x  one  deform  5 . 2 0 show t h e r e s u l t s  VI.  The  The b e n d i n g  roll  i t reaches  strain  , f o r C a s e 1.  Appendix  r e s u l t s a r e as f o l l o w s . increase  A similar  hence  2 5  Bending/Unbending  the bending  x y  i n the p r e s e n t case and  bending.  of bending  peak  i s about  e  f o r t h e c a s e s o f b u l g i n g a l o n e and  Figs.5.19 of  strain,  V shows t h e maximum b u l g i n g d e f l e c t i o n  combination  5.1.3  the  along the inner surface  i n the bulging analysis.  a n d No.2 r o l l s  combination  Fig.5.17,  From  periodically  b e e n r e p o r t e d by M a t s u m i y a .  No.1  the r e s u l t s of the bulging  The m a g n i t u d e o f t h e s h e a r E  Table  show  The r e m a i n i n g  strain,e^,  a steady-state their  results  major c h a r a c t e r i s t i c s  before the bending  about  of c a l c u l a t i o n s  own  is  are of the  observed  point to Roll  level.  The  neutral  upper axes t h e  to No.1 and  E CJ CNJ  \ .  i-v. too.*;  F i g . 5.8  11»  u ' . ' ' • • Vs . -v. •".:•  Sulfur P r i n t of a Longitudinal Section  ; Rating of Internal Cracks -  0.2 46  > 1  E  Fig.  5.9  Sulfur  P r i n t of a L o n g i t u d i n a l  Section  ; Rating  of  I n t e r n a l Cracks  =0.5.  46  Ln  F i g . 5.10  Sulfur P r i n t of a Longitudinal Section ; Rating of Internal Cracks  =1.0. 46 ON  F i g . 5.11  Predicted Bending and Bulging Strain, (Upper S h e l l , V=1.6m/-min)  e  x  l  n  Case 1.  78  Roll  No. at Straightening 0 1  Zone I  2  1  0  c  |-0-5f CP c  R = IO-5m = 471 mm 'R  J?  V = 1-6 m / m i n S = 8 3 mm o =930°C  CD  c o  T  —  c  Inner s u r f a c e Outer s u r f a c e  <u  m  u -2-0  Cose I  1  F i g . 5.12  Predicted Bending and Bulging S t r a i n , (Lower Shell, V=1.6m/min)  e  x  i n Case 1.  79 .  Cose 2  £  1-01  c 'o  R = IO-5m l = 471 m m R  V =1-2 m / m i n S =97mm T = 900°C  I-  CO  0  c  — Inner s u r f o c e - - Outer s u r f a c e  CD c o  0-5  CP C  c OJ  CD  Roll  F i g . 5.13  No. at Straightening  Zone  Predicted Bending and Bulging S t r a i n , e (Upper S h e l l , V=l.2m/min)  i n Case 2.  80  r  Case 3  F i g . 5.14  Predicted Bending and Bulging Strain, (Upper S h e l l , V=l.Om/min)  ^  i n Case 3.  81  F i g . 5.15  Predicted Bending and Bulging Strain, (Lower S h e l l , V=l.Om/min)  i n Case 3.  Fig.  5.16  P r e d i c t e d D i s t o r t i o n Due  to Bulging  i n C a s e 1.  (Lower S h e l l )  00  XX-STRAIN  1 = -0.4257E-03 2 = -0.2556E-03  3 - -0.8544E-04 4 - 0.8471E-04 5 = 6 7 =  8 =  9 = 10 =  Fig. 5.17  0.2549E-03 0.4250E-03  0.5952E-03  0.7653E-03 0.9355E-03 0.1106E-02  Predicted XX-STRAIN Due to Bulging i n Case 1 . (Lower Shell)  XY-STRAIN 1 2 3 4 5 6 7 8 9 10  F i g . 5.18  = = = = = = = = =  -0.4574E-03 -0.3410E-03 -0.2245E-03 -0.1080E-03 0.8423E-05 0.1249E-03 0.2413E-03 0.3578E-03 0.A743E-03 0.5907E-03  Predicted XY-STRAIN Due to Bulging i n Case 1. (Lower S h e l l )  85  Table V  Maximum b u l g i n g d e f l e c t i o n between No.1  Bulging *B CASE 1.  Bending and Bulging (mm)  6  T  U  0.22  0.87  L  0.22  0.39  2.  U  0.13  0.26  3.  TJ  0.11  0.17  L  0.11  0.12  4.  U  0.15  0.40  5.  U  0.13  0.22  6.  U  0.08  0.16  7.  U  8.  U  0.10  0.12  9.  TJ  0.14  0.35  10.  TJ  0.17  0.39  •  and No.2  0.21  TJ : upper s h e l l  (mm)  0.51  ,  L : lower  shell  rolls,  86  l o c a t i o n s of which are roughly center  plane  of  from t h e i n n e r 5.20. and  the slab thickness,  surface  Therefore  lower  center  shells  of each s h e l l  the bending s t r a i n also  plane of s l a b Small  are  peak  can  be a t t r i b u t e d t o  bending  the  peak  move  .further  strain  and  downstream  with  surface;  and  respect  however  and  the  described  at this  strain  the  peak  of  the  of a dynamic  the  effect  of  at  occurrence  to  simplifications, the  the  between  and  bending  accurate  and  owing t o t h e  t h e model of  point  predictions  internal  cracks  earlier.  where  shows t h e p r e d i c t e d  p  was  calculated  curvature,  p  ,  of  the  from t h e r e s u l t s of bending  as f o l l o w s ;  P  in  on  No.-1  this  to decrease  these  be r e a s o n a b l y  was o b t a i n e d  .data  Fig.5.21 strand,  Despite  distributions  that  plant  the  b e n d i n g m o d e l was r u n a n d  l o c a t i o n was f o u n d  i t should  good - a g r e e m e n t  to  approximations  the simulation  a three-stage  downstream.  of  and  d i s t r i b u t i o n s of t h e upper  simplifying  condition  stress  the  respectively  a r e o b s e r v e d between R o l l  at the inner  the  simulation  strain  to  a s shown i n F i g s . 5 . 1 9  p r o c e s s by a t w o - s t a g e b e n d i n g m o d e l . To c h e c k the  respect  7 a n d 14 mm  symmetrical  strains  the tangent(O) r o l l  boundary  with  thickness.  and  upstream  symmetrical  which  =  (  e  2  -  z  ±  )/Ay  (16)  2 =strain  at outer  surface  1 =strain  at inner  surface  e  z  87  Ay =distance between outer and  inner surfaces  As to the question of whether the strand i s straightened the  roll  along  p r o f i l e or not, the r e s u l t s show that bending  occurs  along the curvature determined by the r o l l p r o f i l e as shown Fig.5.21.  A  s i m i l a r f i n d i n g has been reported by O n i s h i  the one-dimensional dynamic a n a l y s i s of bending of  3 1  in for  continuously  cast s l a b s . Fig.5.22 strain  shows  the  in the upper s h e l l and  r e l a t i o n s h i p between  bending  r o l l p i t c h for d i f f e r e n t  surface  temperatures at the straightening which  was  point.  Geometrical  c a l c u l a t e d by assuming a neutral a x i s at the  plane of the slab thickness, also is shown on the as  strain,  a. broken  line  center  same  figure  for comparison( y i s the distance from the  center plane). From the r e s u l t s , i t is evident that the bending strain  i s independent of r o l l p i t c h and  geometrical  strain  can be  on  seen  increases the  the  by 0.05%  stiffness  deformation  is  elongation due  of  by about 0.3%. surface  smaller than the  above  However, a small dependence  temperature;the  bending  strain  with a temperature increase of 90°C because the  strand  localized  is  diminished  so  that  the  around the bending p o i n t . Thus, the  to bending i s enhanced at the s t r a i g h t e n i n g zone  at higher, temperatures. Figs'.5.23 and bending  strain  and  5.24  shell  show  the  thickness  relationship for  the  machine r a d i i . The bending s t r a i n d i s t r i b u t i o n s  10.5m in  between and  the  8.0m upper  88  and  lower  strain in  shells  at  shell  strain  are  the  inner  thickness. at  negative  the  value  seen  be  surface  According  inner i f the  Fig.5.25  symmetrical  changes to  shell  on  the  bending  strain  change  of  curvature.  However,  much  influenced  by  that  neutral  is  not  reason  is  close  to  inner  surface  5.2  the  Corner  the  inner  i s almost  strain  and  to  the  estimate  gained If  at  from  shell  is  Table close  VI to  bending  a  neglected of  neutral  presents the  of of  than  106  mm.  at  the  has  the  a  as  which  predicted  radius.  The  is located  very  on  at  the  inner  at  the  slab  i t is  well,based  can  be  of  the  bending as  surface  stresses  of  the  insight  calculations. i n the  edge  calculated slab  s t r a i n s at well  and  possible on  distribution  center  front  a  surface  strain  focused  of  corner  the  with  .  bending at  The  curvature.  strain  axis  shell.  inner  machine  a  machine  upper  bending  of  the  solidification  strain  have  the  formation  the  can  three-dimensional,elasto-plastic  result  considering  at  bending  strain  therefore  analysis  the  linearly  l o n g i t u d i n a l mid-plane  the  is a  this  increase  increases  axis  independent  an  of  in the  change,  and  crack  strain  bulging  the  bending  shell  influence  surface  the  surface  Although strains  outer  upper  is larger  strain  the  with  results,  the  thickness  bending  the  of  and  linearly  these  surface  shows  radius(curvature) at  to  as shell  by  thickness. the  corner  bulging  and  i n the  mid-  89  Cose i  1.0  R = IO-5m l = 47I mm R  V=l-6m/min S= 8 3 m m T = 930°C 0  x  €2  ty Inner s u r f a c e  'o CO  Ouler surface  0.5  c  TJ C  CD  0  -1 Roll  0  1  No. at Straightening  Zone  ro  F i g . 5.19  €, =0-1  xio_  2  -^ = 084X10  Predicted Bending S t r a i n , e V=1.6m/min)  x  i n Case 1. (Upper S h e l l ,  90  V V ^  I  -2 - 0 21X10 -0-82X10 neutrol axis  No. at- Straightening  0  0|  Zone  1  ty c  2 -Q5  Inner s u r f a c e —  Outer surface.  to CP  c XJ  c o CD  R =10-5 m l = 47lmm  v,  —  -1.0h  T~X.'  ^ \  R  N  V =1-6 m / m i n S = 83mm T =-930°C  /  €2  0  Cose I  L  F i g . 5.20  Predicted Bending Strain, V=1.6m/min)  e x  i n Case 1. (Lower S h e l l ,  1  Roll F i g . 5.21  I  No. at  Straightening  ~1  Zone  Predicted Curvature of the Shell Due to Bending i n Case  7  92  ~U>r  c o (/)  outer inner (Y=1ia9) (Y=40.1)  cn c  •  m  400  471  Roll  F i g . 5.22  850  A  900 990 cxxxTetricd strain  540  Pitch (mm)  Relation between Bending Strain, e and R o l l P i t c h Predicted by the Finite-Element Bending Analysis. x  _J  80  i  i  •  90  100  no  Shell thickness F i g . 5.23  •  •  o  T(«c)  (mm)  Relation between Bending S t r a i n , and S h e l l Thickness Predicted by the Finite-Element Bending A n a l y s i s . (Machine Radius = 10-.5n>)  93  c  outer  (Y=H8S)  inner  •  o  CO  cn c  T Cc) 900  ^metrical strain  TJ C Q)  CQ  90  8 0  100  110  Shell thickness (mm) F i g . 5.24  Relation between Bending Strain, c and S h e l l Thickness Predicted by the Finite-Element Bending Analysis. (Machine Radius = 8.0m) x  machine radius ( m )  13  10.5  -f—  outer  inner  o  •  8  inner inner  T Cc) 900 geometrical strain  (Y=1183) (Y= 52.4) (Y=«X1)(Y=32.2)  •  •  c  a i_  o TJ C at  CO  'V  'A  08  09  1.0 .-^  curvature F i g . 5.25  1.1  12  13  p (Vmm) x10"  4  Relation between Bending S t r a i n , e and Machine Radius (Curvature) Predicted by the Finite-Element Bending Analysis, x  94  plane.  From  the  temperatures internal  results  and small  cracks  in  suppress  the bulging  radius(  Case  internal  cracks  on  the  critical  the  internal  the  the  compared casting  literature  of  1x10"*  strain  the estimated a  strain  an u n d e r e s t i m a t e this  analysis.  partially  stress-strain the bulging  strain  to  bulging,and (based  bow-type  i s obviously owing  on t h e  casters, unfavorable  to  the  low  strain  for internal  previously value  rate  of  cracks  Creep  be  strain  critical  fully  been  have  been  were  increased  to  on value  rate  i s the may  accounted taken  strain  able  in  2.1. I n t h e  based  1  an a p p r o x i m a t e  I f t h e model would  s"  this  effects  at a strain  critical  3xl0~*  creep has not  reported  i n Section  of the  However,  by c o n s i d e r i n g curves.  (Case 1 0 ) ,  scatter(0.2-3.0%  i n slabs. since  0.9m/min(based  1.4m/min  13m  that  form.  1  of cracks  creep  and  machine  radius  due  bending  8.0m  10.5m  e x h i b i t s some  appearance  the  of  of  cracks  critical  at  account  i s nearly  8m  t o ensure  i s  13m m a c h i n e  i n the mid-plane  s" ) as mentioned  study,  in  at the corner  prevent  conditions  an  speed  surface  to  these of  casting  For a  radius  a t which  since case  i n one p o i n t  the values  0.25-0.3%  for  are preferable  the  e f f e c t s on t h e c r i t i c a l  the  be  In  speed  Thus,  machine  speed  occur  2 , 4 , 5 , 6 a n d 7, l o w  mid-plane,  strain.  casting  The  present  the  strain).  cracks  with  5.3 C r e e p  pitches  do n o t a p p e a r  strain).  small  roll  Case  8,9), the threshold  threshold  critical  of  into  rate f o r consider  and as a r e s u l t the  Table  VI  Bending  and B u l g i n g  strain  at s o l i d i f i c a t i o n  corner  center  front.  internal  cracks  center  , corner  U  0.55/0.65  L  0.1  2.  U  0.25/0.3*  0.27  3.  u  0.15/0.2  0.18  no  crack  L  0.1/0.13  -0.18  no  crack  4.  U  0.38/0.4  0.27  center  5.  U  0.15/0.23  0.27  no  crack  6.  U  0.15/0.20  0.27  no  crack  7.  U  0.3/0.4  0.27  center  8.  u  0.15/0.2  0.35  corner  9.  u  0.35/0.4  0.53  center  10.  u  0.3/0.33  0.21  center  CASE 1.  0.4 -0.4  U  : upper  shell  ,  L : lower  no  crack  (^critical  strain)  , corner  shell Ln  96  total  strain  of b e n d i n g and b u l g i n g  Fig.5.26 bulging  strain,  shows t h e t o t a l ,and b e n d i n g  E  should  strain,  E  T  =  be c o n s i d e r e d  (  5)e  2 -  from E q . ( l 7 ) ,  significantly  for  . From t h e r e s u l t s ,  the  ^  bulging  (  strain  affects  the  a n d h e n c e c r e e p e f f e c t s on b u l g i n g  to determine  instance,  +  B  the c r i t i c a l  bulging  strain  strain  strain  of 0 . 1 % which  C a s e 2, t h e c r i t i c a l  strain  will  6 . 6 % , see F i g . 5 . 2 6 .  more  i s increased  0.12% a t the bending  to  of  among t h e s e v a r i a b l e s i s a s f o l l o w s ;  As i s a p p a r e n t  If,  function  e  u  e  strain  increased.  s t r a i n , x ,as a  B  correlation  be  is  be i n c r e a s e d  a  1 7  )  total should  precisely.  from 0.06% t o condition  easily  of  from 0.25%  Bend i ng Fig.  5.26  St r a i n  (%)  P r e d i c t e d T o t a l B e n d i n g and B u l g i n g S t r a i n , , a t an I n n e r S u r f a c e F u n c t i o n o f B u l g i n g S t r a i n , eu, a n d B e n d i n g S t r a i n , E . U  as  a  98  Chapter 6  CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK  6. 1 C o n c l u s i o n s A  two-dimensional  elasto-plastic  developed, based on the p l a n e - s t r e s s calculate  the  bending  s o l i d i f i e d continuously The  and  steel  has  finite-element  bulging  cast  model  been  method  to  deformation of p a r t i a l l y  slabs d u r i n g  straightening.  f i n d i n g s of the study a r e as f o l l o w s ;  (1)  The  model p r e d i c t i o n s of i n t e r n a l  agreement with the p l a n t internal upper  cracks  shell  straightening strain  of  zone  0.25  to  showed good  data of O i t a works, NSC. The  are p r e d i c t e d  beneath  cracks  to occur mainly  the r o l l based  on  i n the  support p o i n t s a  critical  0.3% at a s t r a i n  i n the  cracking  r a t e of 1x10-"  s- . 1  (2)  The upper and lower s h e l l s their  respective  neutral  deform i n d e p e n d e n t l y about  axes.  99  (3)  The r o l l in  friction  the bending  adopted to  for  shift  the  The shell the  is  (5)  y  Bending  of r o l l  neutral  as  axes  a result  a single shells  of  friction.  and hence  bending  beam  i s the distance  by  of e  x  was Owing  the  o f t h e two front  t h e upper  are located  role  0.45  very  roll shells  into and  beam. T h e n e u t r a l  solidification  distribution  the bending  roll  (6)  like  thickness  value  the s o l i d i f i c a t i o n  and  respective  ordinary  where  the  and lower  strain  a  important  e x e r t e d on t h e s t r a n d  past  deform  upper  their  (4)  force  steel,  shells  and  the coefficient  inward,  molten  p l a y s t h e most  analysis  the restraint  friction  force  the lower  axes  of  close . to  fronts.  i s linear  bending  through the  strain,  ,  follows  theory.  from  the neutral  axis  and  R  radius.  occurs along  the curvature determined  by t h e  profile.  The b e n d i n g temperature temperature  strain  depends  of the strand increase  slightly increasing  o f 90°C.  on  the  surface  by 0.05% w i t h  a  100  (7)  The bulging' d e f l e c t i o n  is  enhanced s i g n i f i c a n t l y  as a  c  result  of  bulging the (8)  interaction deflections  lower s h e l l  The  shear  with  bending.  are g r e a t e r  The  i n the  resultant  upper than i n  .  strain,  e x  y  ,  is  comparable  to  the  e  x  component  in  whereas e  the  is  case  of  f a i r l y small,  the close  bulging to one  analysis,  f o u r t h of e  j  X  component, bulging (9)  The  i n the  strain, e  each component u  of  , as  the  T  strain, E  bulging  = (2 - 5 ) e  B  +. c  affects  and hence to prevent suppress  temperatures  straightening  -  However,  and  0.30% it  b  ,  i n terms of and  bending  u  at  the  total  strain  i n t e r n a l cracks  the b u l g i n g by e n s u r i n g  and have a s m a l l r o l l  it low  p i t c h in  zone.  The p r e d i c t e d c r i t i c a l s t r a i n f o r 0.25  bending  , can be e x p r e s s e d  strain  important to  surface  (10)  t  bulging  significantly is  combined  follows.  E  The  the  analysis.  total  strain,e  case of  1x10-" s"  w i l l be necessary  1  for to  i n t e r n a l cracks low-carbon take  into  is  steels. account  101  creep  effects  critical  (11)  In  to  compared  bending,bow-type  radius with  of the  8m  speeds  solidification  front  formation.  is  values  normal c a s t i n g  crack  a more p r e c i s e  value  of  the  strain.  one-point  machine  obtain  of  the  casters,  obviously 10.5m  and  tensile  exceeds  the  a  small  unfavorable 13m b e c a u s e  strain  critical  at  : at  the  value  for  102  6.2  Suggestions for future  work  An i m p o r t a n t d i r e c t i o n f o r f u r t h e r experimental  measurement  of s e v e r a l  strain  investigation the  proposed  for  internal  cracks.  would h e l p c o n c l u s i v e l y  friction  a  the  i n the  and  the  The r e s u l t s o f s u c h  an  e s t a b l i s h the v a l i d i t y  of  model.  T h e n , an o b v i o u s e x t e n s i o n o f t h e w o r k develop  is  parameters adopted  p r e s e n t model s u c h a s t h e c o e f f i c i e n t o f r o l l critical  research  model  for multi  bending c a s t e r s ,  great  i n t e r e s t i n the industry,  point  bending  casters.  based  on  the  would  be  to  which has been of model  for  one  103  REFERENCES  1.  A.Grill,J.K.Brimacombe and F.Weinberg: "Mathematical Analysis of Stresses i n Continuous C a s t i n g of S t e e l " I r o n m a k i n g & S t e e l m a k i n g ,No.1 ,1976, p p 3 8 - 4 7  2.  A . G r i l l a n d K . S o r i m a c h i : "The t h e r m a l l o a d s i n t h e f i n i t e element analysis of e l a s t o - p l a s t i c stresses" Numerical M e t h o d s i n E n g i n e e r i n g , V o l . 1 4 , 1979, p p 4 9 9 ~ 5 0 5 '"  3.  W.T.Lankford : "Some C o n s i d e r a t i o n s o f Strength and D u c t i l i t y i n the Continuous Casting Process" , Metl. 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S c h w e r d t f e g e r : " C o m p u t a t i o n of b u l g i n g of cast slabs with simple bending theory", Steelmaking, No.2 1979, pp68-74  and  Y.Nakamura:  : Kobe-seiko-giho, Stahl  u.  29-3  Eisen,  analysis of cast steel ppl3l-135  Tetsu-to-Hagane  ,1979, 98  advances design",  Analysis Model",  ,Vol.68,  pp55-59  , 1978,  pp254-259  in matrix 1969  in  Engineering  methods  of  106  31.  K.Onishi,K.Nagai and M.Wakabayashi: " A n u m e r i c a l a n a l y s i s of strains i n slabs and forces on rollers in the straightening zone of continuous c a s t i n g machine", Tetsuto-Hagane, Vol.67, 1981,ppl162-1171  32.  J.A.Schey: " Lubrication)",  Metal MARCEL  33.  Suzuki: Trans.  Japan  34.  T.Inoue and H.Tanaka: " Progress i n large-section slab continuous c a s t i n g techniques a t Nippon S t e e l C o r p o r a t i o n " , N i p p o n S t e e l T e c h n i c a l R e p o r t , No.13, J u n e 1979 ,pp1-23  35.  N.Yamauchi,H.Misumi,Y.Uchida and T.Yamamoto: " Internal Cracks i n C o n t i n u o u s l y Cast S l a b s " , Nippon S t e e l T e c h n i c a l R e p o r t , No.13, June 1979, pp62-72  36. -  S . S . D a n i e l :" R o l l C o n t a i n m e n t M o d e l f o r S t r a n d - C a s t and Blooms", 2nd Process Technology Conf.,Chicago, PP102-113  37.  K . N a r i t a , T . M o r i and J . M i y a z a k i : " E f f e c t of Deformation on the Formation of Internal Cracks i n C o n t i n u o u s l y Cast Blooms", Tetsu-to-Hagane, V o l . 6 7 , 1981, p p l 3 0 7 - 1 3 l 6  38.  H.Suzuki: " C h a r a c t e r i s t i c s above 600°C", T e t s u - t o - H a g a n e ,  of Embrittlement in V o l . 6 5 , 1979,S2038  39.  T.Matsumiya:  Vol.69,  40.  T.Obinata : " General View of the Continuous Casting Equipment", Tetsu-to-Hagane, V o l . 6 0 , 1974, pp741-754  41.  F . W e i n b e r g : " The D u c t i l i t y o f C o n t i n u o u s l y C a s t S t e e l Near the M e l t i n g P o i n t - Hot Tearing", Metal.Trans.B, Vol.lOB ,June 1979, p p 2 l 9 - 2 2 7  deformation D E K K E R , New  Institute  Tetsu-to-Hagane,  processes(Friction Y o r k , 1970  of Metals,  and  32, 1968, pp130l  Slab 1981,  Steels  1983, S169  107  42.  H.Suzuki,S.Nishimura and S.Yamaguchi: " C h a r a c t e r i s t i c s of hot d u c t i l i t y in steels subjected to the melting and s o l i d i f i c a t i o n " , Trans.ISIJ, Vol.22, 1982, pp48-56  43.  G.Komma : " D e s i g n and O p e r a t i o n a l A s p e c t s i n C o n t i n u o u s C a s t i n g of Wide S l a b s " , I r o n and Steel Eng., June 1973, pp68-73  44.  A.Vaterlaus straightening 1982,S170  45.  S.Nagata  and  46.  H.Misumi  : Private  :" with  Finite liquid  K.Yasuda:  element analysis for slab core", Tetsu-to-Hagane, Vol.69,  Tetsu-to-Hagane,  Communication  Vol.68,  1982,  S991  108  APPENDIX I  MECHANICAL PROPERTIES ADOPTED IN THE BULGING CALCULATION FOR THE COMPARISON WITH THE EXPERIMENTAL RESULTS OF M o r i t a  Ma t e r i a 1  Material 1  Mater i a l  Ma t e r i a  2  3  4  T  C O  1101  1 205  1 309  1413  E  MPa  17738  1 5827  13916  1 01 23  ay  MPa  14.8  10.1  6.3  2.4  0.36  0.37  0. 38  0.39  a=  75.4 e  °'  2 3  o=  47.Oe ' 0  2 1  <r-  21.5  e ' 0  „ 0.11 6.0 e r  1 6  a =  1  n  109  APPENDIX I I  DERIVATION  OF THE FINITE-ELEMENT  FOR THE E L A S T O - P L A S T I C  Displacements  {  5  {6}  where of  [N  nodal  displacements.  By  all  e a c h e l e m e n t a r e g i v e n by  (A.V)  e  and  displacements  The s t r a i n  f i s a vector  { e } and s t r e s s  { a } i nthe  by  {6}  {e}  = [B]  io]  = [D] ie]  e  (A.2)  applying the p r i n c i p l e  elements,  through  = [N] { 6 }  PROBLEMS  ] i s a m a t r i x o f s h a p e f u n c t i o n s a n d {6  element are given  elements  } within  EQUATIONS  summing  of v i r t u a l  the i n d i v i d u a l  Eq.(A.3)  is  equilibrium  obtained.  and t h e d i s t r i b u t e d  work  The  loads { } p  to  these  equations f o r  nodal  c a n nov? be  forces, related  110  i F  }= [ K ] {  } + { Fp' }  where  [ K ) = Z / [ B ]  and  {• Fp }=  T  I "N J  -If  (A.3)  l D ] l B ] d T  (vol)  { P } d (vol)  E - summation over a l l elements. Equation  ( A . 3 ) c a n be  solved  f o r the  { 6 } = [ K J  Finally, stress  using  distribution  Under analysis  i s  strain(  solved  by  incrementally  The  increment  remaining  load  [K ]  i s divided  yield.  [  ]  matrices  {  }  vectors  of the d e r i v a t i o n s  and  2 9  .  are  the  the [D ] and  The  i s  adjusted  discussed  matrices  i s  such  i s  applied  after  a l l elements equal  stress  Eq.(A.3)  load  are adjusted, After  [ K ]  Therefore  2 9  into several  ** d e t a i l s Yamada  deformation  method. matrix  and  region.  since  increments  (A.4)  can c a l c u l a t e the s t r a i n  dependent.  incremental  30 e l e m e n t s  load  Nomenclature;  one  of p l a s t i c  stress)  and t h e  C { F } - { Fp } )  the e n t i r e  conditions  or  an  increment.  over  ,  more c o m p l i c a t e d  become  each  Eq.(A.2)  - 1  displacements.  every  that  with  yield,  the  increments.  by  Zienkiewicz  2 8  Ill  APPENDIX I I I  MATERIAL M A T R I X [ D ] ( p l a n e  Below the plane  the  s t r e s s ) USED I N THE F I N I T E ELEMENT  yield  stress condition  point,  the e l a s t i c  2  I.v  where E i s t h e Young's Modulus  material  [D ] P  which  was  0  (B. 1 )  1-y  sym  matrix  e  0 1  conditions  [D ] for  i s g i v e n by  1 - v  Under  matrix  and v i s P o i s s o n ' s  of p l a s t i c developed  ratio.  deformation, the by  Yamada  plastic  f o r a Mises  1 9  i s as f o l l o w s .  I  DP ]  1 -  1  sym  v  0  1  0  S  1 2 S  s  1-v sym  S  2  S  1 6  S  2 6  S  S  (B.2)  112  W  h  e  r  S = j a % . + S.a;  e  S a  n  d  +  ( <r' + vo  l = 1~2  x  y  s/  t h e o, eP c u r v e . o '., o t a n d x x  Unloading as  y  2  check  x y  2S x'xy  +  ). S -  "a a n d 7 P a r e t h e e f f e c t i v e s t r e s s of  y  6  ^  va' + a ; ) . S,x  and s t r a i n  are  a n d H* i s t h e s l o p e  deviatoric  h a s been p e r f o r m e d  stresses.  by c a l c u l a t i n g  de  p  follows.  d  £P  S,d e + S.de + S,dy 1 x 2 y 6 lxy  ,_ , , (B.3)  3 J_ 2 o"  i f de  Once  <0  p  unloading  changed from  unloading  occurs,  the material  [D ] to [ D ] . p  e  matrix  of the element i s  113  APPENDIX  IV  THICK-WALLED CYLINDER UNDER INTERNAL P R E S S U R E ( p l a n e  To performed  for  internal stress is  field  the  the  pressure  shown  r=c.  check  case for  of  The  displacements  of  compare the  numerical  Hill  3  0  .  the  Mechanical  Poisson's  ratio  a  cylinder results  properties  to  with of  Young's modulus  E=10.4X10  6  the  the  v-0.3 psi 6  psi  cylinder solution  the  axial as  the  at  radial  symmetry.  solution  were as  the  a<r<b  b=2a, i n o r d e r  analytical  material  of  i s located  on  was  under  cylinder  direction  the  calculation  boundary  were s e l e c t e d  G=4xl0  stress  6  the due  a  analytical  elasto-plastic in  ?!  thick-walled  g e o m e t r y of  Shear modulus  Yield  "EPIC-IV  an  The  b o u n d a r i e s were c o n s t r a i n e d dimensions  of  which  is available.  in Fig.IV.1.  The  accuracy  strain)  The to of  follows:  114  The  stress  be o b t a i n e d f r o m the those  comparison of H i l l .  ^ d e p e n d s on t h e s t r a i n z  the Prandtl-Reuss of t h e c a l c u l a t e d  Excellent  agreement  equations. cr z  h i s t o r y and Fig.IV.2  i n dimensionless  i s observed.  form  must shows with  115  F i g . IV.1  Geometry of a Thick Walled Cylinder,  0.4  F i g . IV.2  Comparison of the Calculated Stresses °z: Csolid points and l i n e s ) with Those Obtained by H i l l " , 3 0  1 17  APPENDIX V  ESTIMATION OF ROLL FRICTION  R o l l No. R o l l  FORCE IN CASE 1(UPPER SHELL)  P i t c h R o l l F r i c t i o n Cumulative Roll Friction Force Force H K mm i F i Fi  Stress  Average Strain  MPa o  min  % e i  i  Elongation  .At. i  0  0  0  6.4  0.008  0.03  12.9  0.02  0.08  406.6  19.6  0.03  0.13  141.1  546.7  26.3  0.035  0.15  435  143.1  689.8  33.2  0.06  0.26  5  435  143.1  832.9  40.1  0.095  0.41  • 4  435  148.9  981.8*  47.3  0.15  0.65  3  471  157.7  1139.5  54.9  0.32  1.51  2  471  157.7  1297.2  62.5  0.48  2.28  1  471  157.7  1454.9  70.1  0.71  3.36  Total  8.86  11  435  133.2  0  10  435  136.2  133.2  9  435  137.2  269.4  8  435  140.1  7  435  6  (B.P.)  * Fu » 981.8 CFig.3. 13)  N i s adopted as the force boundary condition on upstream edge.  APPENDIX V I  RESULTS OF CALCULATION OF BENDING (CASE 2 TO CASE 10)  AND  BULGING  119  Cose 2  1.0  R = IO-5m ! =47lmm R  V =l-2 m / m i n S =97 mm T =900°C  V \  €2  c Inner s u r f a c e  cn  Outer  surface  a. 0.5 c  '•XD  c CD  0  F i g . VI.1  _1  0  Roll  No. at- Straightening  Predicted Bending S t r a i n ,  1  e  x  Zone  i n Case 2 . (Upper Shell)  120  r Cose 3  1.0  R =IO-5m ' =4 7 I mm R  V = IO m / m i n s = I 0 6 mm 900°C  V  1  ^7  ty  €2  c "o  Inner s u r f o c e Outer  CO  c  »J.  surfoce  0.5  C  CD  Oi  -1 Roll  F i g . VI.2  Predicted  0  ^1  No. at- Straightening  Sending Strain,  i  n  Zone  Case.3. (Upper Shell)  121  €,=-0-02 X I O " € = ^0-85X10"  Roll  A  cn  2  No. at- Straightening 0  Zone  1  2  x  .E -0.5|  Inner Ouler  O  surfoce surfoce  CO cr> c  R=IO-5m I =471 m m  c  OJ  R  ca -1.0  €2  V =I0 m/min S =106 m m T = 900°C  Cose 3  F i g , VI.3  Predicted Bending Strain,  c  x  i n Case 3.  (Lower Shell)  Roll F i g . VI.4  No. at Straightening Zone  Predicted Curvature of the Shell Due  to Bending i n Case 3.  1  1  R= 10-5 m 1 = 471 mm R V=l-2m/min S = 97 mm T= 900°C  1.0  r "*  /  1 1  Inner surfoce Outer surfoce  c o CO  r  Cose 4  *  / /  0  ty  1  1  0.5  c c CD  o  1  -1  0  1  Roll  No. at- Straightening  Zone  neutral axis LO  €, =oi  xio_  2  0-91 XI0  Fig.  VI.5  Predicted Bending S t r a i n , e . i n Case 4, x  (Upper Shell)  124  r  Cose 4  R = IO-5m ?= 471 mm  10  V = 1-2 m / m i n S = 97 m m  T = 990°C 0  c o  Inner s u r f a c e Outer s u r f o c e  CO CP  c '^0  5  CD TJ  c D  cn c TJ C O)  CD  Roll  F i g . VI.6  No. at  Straightening  Zone  Predicted Bending and Bulging S t r a i n , e (Upper Shell)  x  i n Case 4.  125  T Cose 5  R= tO 5 m I = 471 m m  1.0  V = l-2m/min S= 97 m m  T^= 900°C  -^--'-v €2  c 'o  Inner s u r f o c e  00  Ouler  surfoce  CV OS cz o CD  o  l  -1  0  Roll  No.  CO  CNJ  F i g . VI.7  1  at- Straightening  Zone  € =003XIC[J 0:79X10  Predicted Bending S t r a i n , ^  i n Case 5, (Upper Shell)  126  Cose 5  ioh  R =10 5 m l = 47lmm R  V = 1-2 m / m i n S = 9 7 mm T =850°C 0  c "o  Inner s u r f a c e Outer s u r f a c e  CO CP  c  •fO-5 3 CO  -o c o c c cu CD  -  Roll  Fig.  VI.8  1  .  0  No. at Straightening  I Zone  Predicted Bending and Bulging Strain, (Upper Shell)  i n Case 5 .  127  Cose 6  R= I O - 5 m I =400mm  1.0  R  V=l-2m/rnin S= 9 7 m m T= 900°C-  _ I— _ — _  0  '  k-  c I.  CO  —  Inner  —  Outer  surfoce surfoce  c c CD  GQ  0  -1 Roll  0 No. at  OD rsi  F i g . VI.9  1  Straightening  €  Zone  =0-09X10 2  €2= 0 8 4 X 1 0  Predicted Bending S t r a i n , £  x  i n Case 6. (Upper Shell)  128  r Cose 6  R = tO-5 m l = 4 0 0 mm R  V=l 2 m/min S =97mm T = 900°C 0  /  c o  Inner s u r f a c e  CO  Outer surface  \  V  /  €2  CP  c  CP  3 0-5  CO TJ  c D  CP C  TJ  c  OJ  CD  -I Roll  F i g . VI.10  0 No. ot Straightening  I Zone  Predicted Bending and Bulging S t r a i n , (Upper Shell)  i n Case 6.  129  1  r-  ••"' 1  1 Cose 7  R = 10-5 m l = 5 4 0 mm  1.0  R  V = 1-2 m / m i n T =900°C  /  0  /  '  \  **  S = 97mm  -  —  —  f—«T  \'  /  c 'o  -  ~  \ €2  .  ei  -  s  — —  Inner s u r f o c e Outer  /  surfoce  t  t  |» 0.5 0)  1  /  CO  T J C  ~~>  1  CO.  1  1 1 t t 1  /  /  ^  '  /  f  1  0 No.  F i g . VI.11  1  at- Straightening  Predicted Bending Strain,  t  Zone  i n Case 7. (Upper Shell)  130  I Cose 7 -\  0  R = 10-5 m ' = 5 4 0 mm  A  R  V = 1-2 m / m i n S = 97mm 900°C  c  'o  / /  /  \ \  \  V  CO cn c  e2  Inner s u r f o c e Outer s u r f o c e  m 0-5 •o c o cn c  •o c a> CD  0  Roll  F i g . VI.12  No. at Straightening  Zone  Predicted Bending and Bulging S t r a i n , (Upper Shell)  i n Case 7,  131  R = 80m l = 47I mm V =0-9m/min S =97 mm T = 900°C  1.0  R  el  Q  c 'o  cn c T> c  Inner surfoce Outer surface  0.5h  OJ  m  0'  Cose 8  F i g . VI.13  -1  0  Roil  No. at- Straightening  Predicted Bending S t r a i n , e  ^1  x  Zone  i n Case 8. (Upper Shell)  R = 80m I =471 mm  10  n  V = 0-9 m/min S = 97 mm T = 900°C Q  c o  •Inner surfoce Outer surface  CO C P  c  f  0-51  X)  • c o CP  c  TJ  c cu CQ  0  Roll  F i g . VI.14  No. at Straightening  Zone  Predicted Bending and Bulging S t r a i n , (Upper Shell)  i n Case 8.  133  Cose 9  10  R =8 0 m ' 471 mm V =1-2 m / m i n =  R  S = 83mm 900°C  V X  Vl/  Inner s u r f o c e  o CO  Outer  surfoce  0-5  CP C TJ C  CU  CO  0'  Roll  No. at- Straightening  Zone  €, =0-l9Xl6~f  2  £,= 1 0 5 X 1 0  Fig,  VI.15  Predicted Bending Strain,  e  x  i n Case 9. (Upper Shell)  134  — r Cose 9  R =80m ^ =471 mm V = l-.2m/min S =83mm T = 900°C 0  c  10 Inner s u r f o c e Outer s u r f o c e  o  l €2  co CP  c  CD •o 0 5 cr o CP  c c cu CD 0  Roll  F i g . VI.16  No. ot Straightening  Zone  Predicted Bending and Bulging S t r a i n , (Upper Shell)  i n Case 9.  135  r~ Cose  10  R =l30m L^ATImm V = l-47m/min S =97mm T =900°C  ty  0  X  c o  \  €2  CO  — Inner s u r f o c e • - Outer s u r f o c e  "D C OJ  m  Roll  F i g . VI.17  No. at Straightening  Predicted Bending S t r a i n , e  Zone  x  i n Case 10. (Upper Shell)  136  , —  Cose 10  J  R=l30m l =47lmm  10  R  V = l-47m/min S = 97mm T =900°C 0  ' o  /  00  -  Bulgin  O  'cn  Inner s u r f a c e Outer surfoce  /  1 1 1 1 1 1 1  \  \  €2  TJ c o  cn c  TJ  c cu CQ  -I Roll  F i g . VI.18  No. at  0 Straightening  I  Zone  Predicted Bending and Bulging Strain, e (Upper Shell)  x  i n Case 10.  

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